% Mizar problem: t4_bvfunc_2,bvfunc_2,542,28 
fof(t4_bvfunc_2,conjecture,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ( r2_hidden(C,B)
               => r1_setfam_1(C,k3_bvfunc_2(A,B)) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_eqrel_1,cc2_eqrel_1,commutativity_k3_xboole_0,d1_partit1,d2_bvfunc_2,d2_setfam_1,d3_bvfunc_2,d3_tarski,d3_xboole_0,d4_tarski,d6_eqrel_1,d7_xboole_0,dt_k1_bvfunc_2,dt_k1_partit1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k3_bvfunc_2,dt_k3_pua2mss1,dt_k3_tarski,dt_k3_xboole_0,dt_k5_setfam_1,dt_m1_eqrel_1,dt_m1_subset_1,dt_m1_t_1topsp,existence_m1_eqrel_1,existence_m1_subset_1,existence_m1_t_1topsp,fc1_margrel1,fc1_partit1,fc1_subset_1,fc1_xboole_0,idempotence_k3_xboole_0,rc1_margrel1,rc1_subset_1,rc1_t_1topsp,rc1_xboole_0,rc2_subset_1,rc2_t_1topsp,rc2_xboole_0,rc3_t_1topsp,redefinition_k1_bvfunc_2,redefinition_k5_setfam_1,reflexivity_r1_setfam_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,t1_subset,t2_boole,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,t92_zfmisc_1]),
    [file(bvfunc_2,t4_bvfunc_2)]).

fof(antisymmetry_r2_hidden,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    []).

fof(cc1_eqrel_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ~ v1_xboole_0(B) ) ) ),
    file(eqrel_1,cc1_eqrel_1),
    []).

fof(cc2_eqrel_1,axiom,(
    ! [A,B] : 
      ( m1_eqrel_1(B,A)
     => v1_setfam_1(B) ) ),
    file(eqrel_1,cc2_eqrel_1),
    []).

fof(commutativity_k3_xboole_0,axiom,(
    ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ),
    file(xboole_0,k3_xboole_0),
    []).

fof(d1_partit1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( r1_partit1(A,B,C)
            <=> ? [D] : 
                  ( r1_tarski(D,B)
                  & D != k1_xboole_0
                  & C = k3_tarski(D) ) ) ) ) ),
    file(partit1,d1_partit1),
    []).

fof(d2_bvfunc_2,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
         => ! [C] : 
              ( r1_bvfunc_2(A,B,C)
            <=> ( ! [D] : 
                    ( m1_eqrel_1(D,A)
                   => ( r2_hidden(D,B)
                     => r1_partit1(A,D,C) ) )
                & ! [D] : 
                    ( ( r1_tarski(D,C)
                      & ! [E] : 
                          ( m1_eqrel_1(E,A)
                         => ( r2_hidden(E,B)
                           => r1_partit1(A,E,D) ) ) )
                   => D = C ) ) ) ) ) ),
    file(bvfunc_2,d2_bvfunc_2),
    []).

fof(d2_setfam_1,axiom,(
    ! [A,B] : 
      ( r1_setfam_1(A,B)
    <=> ! [C] : ~ ( r2_hidden(C,A)
          & ! [D] : ~ ( r2_hidden(D,B)
              & r1_tarski(C,D) ) ) ) ),
    file(setfam_1,d2_setfam_1),
    []).

fof(d3_bvfunc_2,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ( ( B != k1_xboole_0
                 => ( C = k3_bvfunc_2(A,B)
                  <=> ! [D] : 
                        ( r2_hidden(D,C)
                      <=> r1_bvfunc_2(A,B,D) ) ) )
                & ( B = k1_xboole_0
                 => ( C = k3_bvfunc_2(A,B)
                  <=> C = k3_pua2mss1(A) ) ) ) ) ) ) ),
    file(bvfunc_2,d3_bvfunc_2),
    []).

fof(d3_tarski,axiom,(
    ! [A,B] : 
      ( r1_tarski(A,B)
    <=> ! [C] : 
          ( r2_hidden(C,A)
         => r2_hidden(C,B) ) ) ),
    file(tarski,d3_tarski),
    []).

fof(d3_xboole_0,axiom,(
    ! [A,B,C] : 
      ( C = k3_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d3_xboole_0),
    []).

fof(d4_tarski,axiom,(
    ! [A,B] : 
      ( B = k3_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> ? [D] : 
              ( r2_hidden(C,D)
              & r2_hidden(D,A) ) ) ) ),
    file(tarski,d4_tarski),
    []).

fof(d6_eqrel_1,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( ( A != k1_xboole_0
         => ( m1_eqrel_1(B,A)
          <=> ( k5_setfam_1(A,B) = A
              & ! [C] : 
                  ( m1_subset_1(C,k1_zfmisc_1(A))
                 => ( r2_hidden(C,B)
                   => ( C != k1_xboole_0
                      & ! [D] : 
                          ( m1_subset_1(D,k1_zfmisc_1(A))
                         => ~ ( r2_hidden(D,B)
                              & C != D
                              & ~ r1_xboole_0(C,D) ) ) ) ) ) ) ) )
        & ( A = k1_xboole_0
         => ( m1_eqrel_1(B,A)
          <=> B = k1_xboole_0 ) ) ) ) ),
    file(eqrel_1,d6_eqrel_1),
    []).

fof(d7_xboole_0,axiom,(
    ! [A,B] : 
      ( r1_xboole_0(A,B)
    <=> k3_xboole_0(A,B) = k1_xboole_0 ) ),
    file(xboole_0,d7_xboole_0),
    []).

fof(dt_k1_bvfunc_2,axiom,(
    ! [A] : 
      ( v1_t_1topsp(k1_bvfunc_2(A),A)
      & m1_t_1topsp(k1_bvfunc_2(A),A) ) ),
    file(bvfunc_2,k1_bvfunc_2),
    []).

fof(dt_k1_partit1,axiom,(
    $true ),
    file(partit1,k1_partit1),
    []).

fof(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    []).

fof(dt_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_1),
    []).

fof(dt_k3_bvfunc_2,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A))) )
     => m1_eqrel_1(k3_bvfunc_2(A,B),A) ) ),
    file(bvfunc_2,k3_bvfunc_2),
    []).

fof(dt_k3_pua2mss1,axiom,(
    ! [A] : m1_eqrel_1(k3_pua2mss1(A),A) ),
    file(pua2mss1,k3_pua2mss1),
    []).

fof(dt_k3_tarski,axiom,(
    $true ),
    file(tarski,k3_tarski),
    []).

fof(dt_k3_xboole_0,axiom,(
    $true ),
    file(xboole_0,k3_xboole_0),
    []).

fof(dt_k5_setfam_1,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => m1_subset_1(k5_setfam_1(A,B),k1_zfmisc_1(A)) ) ),
    file(setfam_1,k5_setfam_1),
    []).

fof(dt_m1_eqrel_1,axiom,(
    ! [A,B] : 
      ( m1_eqrel_1(B,A)
     => m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ),
    file(eqrel_1,m1_eqrel_1),
    []).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    []).

fof(dt_m1_t_1topsp,axiom,(
    $true ),
    file(t_1topsp,m1_t_1topsp),
    []).

fof(existence_m1_eqrel_1,axiom,(
    ! [A] : 
    ? [B] : m1_eqrel_1(B,A) ),
    file(eqrel_1,m1_eqrel_1),
    []).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    []).

fof(existence_m1_t_1topsp,axiom,(
    ! [A] : 
    ? [B] : m1_t_1topsp(B,A) ),
    file(t_1topsp,m1_t_1topsp),
    []).

fof(fc1_margrel1,axiom,
    ( v1_xboole_0(k1_xboole_0)
    & v1_margrel1(k1_xboole_0) ),
    file(margrel1,fc1_margrel1),
    []).

fof(fc1_partit1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_partit1(A)) ),
    file(partit1,fc1_partit1),
    []).

fof(fc1_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ),
    file(subset_1,fc1_subset_1),
    []).

fof(fc1_xboole_0,axiom,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    []).

fof(idempotence_k3_xboole_0,axiom,(
    ! [A,B] : k3_xboole_0(A,A) = A ),
    file(xboole_0,k3_xboole_0),
    []).

fof(rc1_margrel1,axiom,(
    ? [A] : v1_margrel1(A) ),
    file(margrel1,rc1_margrel1),
    []).

fof(rc1_subset_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B) ) ) ),
    file(subset_1,rc1_subset_1),
    []).

fof(rc1_t_1topsp,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_t_1topsp(B,A)
      & v1_t_1topsp(B,A) ) ),
    file(t_1topsp,rc1_t_1topsp),
    []).

fof(rc1_xboole_0,axiom,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    []).

fof(rc2_subset_1,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & v1_xboole_0(B) ) ),
    file(subset_1,rc2_subset_1),
    []).

fof(rc2_t_1topsp,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_eqrel_1(B,A)
          & ~ v1_xboole_0(B)
          & v1_setfam_1(B) ) ) ),
    file(t_1topsp,rc2_t_1topsp),
    []).

fof(rc2_xboole_0,axiom,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    []).

fof(rc3_t_1topsp,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_t_1topsp(B,A)
      & ~ v1_xboole_0(B)
      & v1_t_1topsp(B,A) ) ),
    file(t_1topsp,rc3_t_1topsp),
    []).

fof(redefinition_k1_bvfunc_2,axiom,(
    ! [A] : k1_bvfunc_2(A) = k1_partit1(A) ),
    file(bvfunc_2,k1_bvfunc_2),
    []).

fof(redefinition_k5_setfam_1,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => k5_setfam_1(A,B) = k3_tarski(B) ) ),
    file(setfam_1,k5_setfam_1),
    []).

fof(reflexivity_r1_setfam_1,axiom,(
    ! [A,B] : r1_setfam_1(A,A) ),
    file(setfam_1,r1_setfam_1),
    []).

fof(reflexivity_r1_tarski,axiom,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    []).

fof(symmetry_r1_xboole_0,axiom,(
    ! [A,B] : 
      ( r1_xboole_0(A,B)
     => r1_xboole_0(B,A) ) ),
    file(xboole_0,r1_xboole_0),
    []).

fof(t1_subset,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    []).

fof(t2_boole,axiom,(
    ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ),
    file(boole,t2_boole),
    []).

fof(t2_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    []).

fof(t3_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,k1_zfmisc_1(B))
    <=> r1_tarski(A,B) ) ),
    file(subset,t3_subset),
    []).

fof(t4_subset,axiom,(
    ! [A,B,C] : 
      ( ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C)) )
     => m1_subset_1(A,C) ) ),
    file(subset,t4_subset),
    []).

fof(t5_subset,axiom,(
    ! [A,B,C] : ~ ( r2_hidden(A,B)
      & m1_subset_1(B,k1_zfmisc_1(C))
      & v1_xboole_0(C) ) ),
    file(subset,t5_subset),
    []).

fof(t6_boole,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    []).

fof(t7_boole,axiom,(
    ! [A,B] : ~ ( r2_hidden(A,B)
      & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    []).

fof(t8_boole,axiom,(
    ! [A,B] : ~ ( v1_xboole_0(A)
      & A != B
      & v1_xboole_0(B) ) ),
    file(boole,t8_boole),
    []).

fof(t92_zfmisc_1,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => r1_tarski(A,k3_tarski(B)) ) ),
    file(zfmisc_1,t92_zfmisc_1),
    []).