## TPTP Problem File: SEV490^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEV490^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : INJECTIVE_ON_PREIMAGE
% Version  : Especial.
% English  :

% Refs     : [Kal16] Kalisyk (2016), Email to Geoff Sutcliffe
% Source   : [Kal16]
% Names    : INJECTIVE_ON_PREIMAGE_.p [Kal16]

% Status   : Theorem
% Rating   : 1.00 v7.2.0, 0.75 v7.1.0
% Syntax   : Number of formulae    :   15 (   0 unit;   7 type;   0 defn)
%            Number of atoms       :  135 (  13 equality;  93 variable)
%            Maximal formula depth :   22 (   9 average)
%            Number of connectives :  102 (   1   ~;   1   |;   5   &;  93   @)
%                                         (   0 <=>;   2  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   37 (  37   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    9 (   7   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   :   47 (   0 sgn;  32   !;   4   ?;   3   ^)
%                                         (  47   :;   8  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH1_THM_EQU_NAR

% Comments : Exported from core HOL Light.
%------------------------------------------------------------------------------
thf('thf_const_const/sets/SUBSET',type,(
'const/sets/SUBSET':
!>[A: \$tType] :
( ( A > \$o ) > ( A > \$o ) > \$o ) )).

thf('thf_const_const/sets/SETSPEC',type,(
'const/sets/SETSPEC':
!>[A: \$tType] :
( A > \$o > A > \$o ) )).

thf('thf_const_const/sets/INSERT',type,(
'const/sets/INSERT':
!>[A: \$tType] :
( A > ( A > \$o ) > A > \$o ) )).

thf('thf_const_const/sets/IN',type,(
'const/sets/IN':
!>[A: \$tType] :
( A > ( A > \$o ) > \$o ) )).

thf('thf_const_const/sets/IMAGE',type,(
'const/sets/IMAGE':
!>[A: \$tType,B: \$tType] :
( ( A > B ) > ( A > \$o ) > B > \$o ) )).

thf('thf_const_const/sets/GSPEC',type,(
'const/sets/GSPEC':
!>[A: \$tType] :
( ( A > \$o ) > A > \$o ) )).

thf('thf_const_const/sets/EMPTY',type,(
'const/sets/EMPTY':
!>[A: \$tType] :
( A > \$o ) )).

thf('thm/sets/IN_',axiom,(
! [A: \$tType,P: A > \$o,A0: A] :
( ( 'const/sets/IN' @ A @ A0 @ P )
= ( P @ A0 ) ) )).

thf('thm/sets/IN_ELIM_THM_1',axiom,(
! [A: \$tType,A0: A > \$o,A1: A] :
( ( 'const/sets/IN' @ A @ A1
@ ( 'const/sets/GSPEC' @ A
@ ^ [A2: A] :
? [A3: A] :
( 'const/sets/SETSPEC' @ A @ A2 @ ( A0 @ A3 ) @ A3 ) ) )
= ( A0 @ A1 ) ) )).

thf('thm/sets/IN_IMAGE_',axiom,(
! [B: \$tType,A: \$tType,A0: B,A1: A > \$o,A2: A > B] :
( ( 'const/sets/IN' @ B @ A0 @ ( 'const/sets/IMAGE' @ A @ B @ A2 @ A1 ) )
= ( ? [A3: A] :
( ( A0
= ( A2 @ A3 ) )
& ( 'const/sets/IN' @ A @ A3 @ A1 ) ) ) ) )).

thf('thm/sets/EXTENSION_',axiom,(
! [A: \$tType,A0: A > \$o,A1: A > \$o] :
( ( A0 = A1 )
= ( ! [A2: A] :
( ( 'const/sets/IN' @ A @ A2 @ A0 )
= ( 'const/sets/IN' @ A @ A2 @ A1 ) ) ) ) )).

thf('thm/sets/SUBSET_',axiom,(
! [A: \$tType,A0: A > \$o,A1: A > \$o] :
( ( 'const/sets/SUBSET' @ A @ A0 @ A1 )
= ( ! [A2: A] :
( ( 'const/sets/IN' @ A @ A2 @ A0 )
=> ( 'const/sets/IN' @ A @ A2 @ A1 ) ) ) ) )).

thf('thm/sets/NOT_IN_EMPTY_',axiom,(
! [A: \$tType,A0: A] :
~ ( 'const/sets/IN' @ A @ A0 @ ( 'const/sets/EMPTY' @ A ) ) )).

thf('thm/sets/IN_INSERT_',axiom,(
! [A: \$tType,A0: A,A1: A,A2: A > \$o] :
( ( 'const/sets/IN' @ A @ A0 @ ( 'const/sets/INSERT' @ A @ A1 @ A2 ) )
= ( ( A0 = A1 )
| ( 'const/sets/IN' @ A @ A0 @ A2 ) ) ) )).

thf('thm/sets/INJECTIVE_ON_PREIMAGE_',conjecture,(
! [B: \$tType,A: \$tType,A0: A > B,A1: A > \$o,A2: B > \$o] :
( ( ! [A3: B > \$o,A4: B > \$o] :
( ( ( 'const/sets/SUBSET' @ B @ A3 @ A2 )
& ( 'const/sets/SUBSET' @ B @ A4 @ A2 )
& ( ( 'const/sets/GSPEC' @ A
@ ^ [A5: A] :
? [A6: A] :
( 'const/sets/SETSPEC' @ A @ A5
@ ( ( 'const/sets/IN' @ A @ A6 @ A1 )
& ( 'const/sets/IN' @ B @ ( A0 @ A6 ) @ A3 ) )
@ A6 ) )
= ( 'const/sets/GSPEC' @ A
@ ^ [A5: A] :
? [A6: A] :
( 'const/sets/SETSPEC' @ A @ A5
@ ( ( 'const/sets/IN' @ A @ A6 @ A1 )
& ( 'const/sets/IN' @ B @ ( A0 @ A6 ) @ A4 ) )
@ A6 ) ) ) )
=> ( A3 = A4 ) ) )
= ( 'const/sets/SUBSET' @ B @ A2 @ ( 'const/sets/IMAGE' @ A @ B @ A0 @ A1 ) ) ) )).

%------------------------------------------------------------------------------
```