% Mizar problem: t51_sysrel,sysrel,987,46 
fof(t51_sysrel,conjecture,(
    ! [A] : 
      ( v1_relat_1(A)
     => ( r1_tarski(k1_relat_1(k1_sysrel(A)),k1_relat_1(A))
        & r1_tarski(k2_relat_1(k1_sysrel(A)),k2_relat_1(A))
        & r1_tarski(k2_relat_1(k1_sysrel(A)),k1_relat_1(A))
        & r1_tarski(k1_relat_1(k1_sysrel(A)),k2_relat_1(A)) ) ) ),
    inference(mizar_bg_added,[status(thm)],[commutativity_k3_xboole_0,d1_sysrel,dt_k1_relat_1,dt_k1_sysrel,dt_k1_zfmisc_1,dt_k2_relat_1,dt_k3_xboole_0,dt_k6_relat_1,dt_m1_subset_1,existence_m1_subset_1,fc1_relat_1,idempotence_k3_xboole_0,reflexivity_r1_tarski,t25_relat_1,t3_subset,t43_sysrel,t45_sysrel]),
    [file(sysrel,t51_sysrel)]).

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_sysrel,axiom,(
    ! [A] : 
      ( v1_relat_1(A)
     => k1_sysrel(A) = k3_xboole_0(A,k6_relat_1(k1_relat_1(A))) ) ),
    file(sysrel,d1_sysrel),
    []).

fof(dt_k1_relat_1,axiom,(
    $true ),
    file(relat_1,k1_relat_1),
    []).

fof(dt_k1_sysrel,axiom,(
    ! [A] : 
      ( v1_relat_1(A)
     => v1_relat_1(k1_sysrel(A)) ) ),
    file(sysrel,k1_sysrel),
    []).

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

fof(dt_k2_relat_1,axiom,(
    $true ),
    file(relat_1,k2_relat_1),
    []).

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

fof(dt_k6_relat_1,axiom,(
    ! [A] : v1_relat_1(k6_relat_1(A)) ),
    file(relat_1,k6_relat_1),
    []).

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

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

fof(fc1_relat_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => v1_relat_1(k3_xboole_0(A,B)) ) ),
    file(relat_1,fc1_relat_1),
    []).

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

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

fof(t25_relat_1,axiom,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => ( r1_tarski(A,B)
           => ( r1_tarski(k1_relat_1(A),k1_relat_1(B))
              & r1_tarski(k2_relat_1(A),k2_relat_1(B)) ) ) ) ) ),
    file(relat_1,t25_relat_1),
    []).

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

fof(t43_sysrel,axiom,(
    ! [A] : 
      ( v1_relat_1(A)
     => ( r1_tarski(k1_sysrel(A),A)
        & r1_tarski(k1_sysrel(A),k6_relat_1(k1_relat_1(A))) ) ) ),
    file(sysrel,t43_sysrel),
    []).

fof(t45_sysrel,axiom,(
    ! [A] : 
      ( v1_relat_1(A)
     => k1_relat_1(k1_sysrel(A)) = k2_relat_1(k1_sysrel(A)) ) ),
    file(sysrel,t45_sysrel),
    []).