## TPTP Problem File: SEU795^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU795^2 : TPTP v8.0.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : More about Functions - Images of Functions
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! f:i>i.! x:i.in x (image1 A (^ y:i.f y)) <->
%            (? y:i.in y A & x = f y))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC297l [Bro08]

% Status   : Theorem
% Rating   : 0.82 v7.5.0, 0.86 v7.4.0, 0.44 v7.2.0, 0.50 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.57 v6.1.0, 0.43 v6.0.0, 0.57 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 0.80 v4.1.0, 1.00 v3.7.0
% Syntax   : Number of formulae    :   13 (   5 unt;   7 typ;   5 def)
%            Number of atoms       :   28 (  11 equ;   0 cnn)
%            Maximal formula atoms :    6 (   4 avg)
%            Number of connectives :   46 (   0   ~;   0   |;   4   &;  31   @)
%                                         (   4 <=>;   7  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   12 (   3 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :   14 (  14   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    8 (   7 usr;   3 con; 0-2 aty)
%            Number of variables   :   25 (   8   ^  13   !;   4   ?;  25   :)
% SPC      : TH0_THM_EQU_NAR

%------------------------------------------------------------------------------
thf(in_type,type,
in: \$i > \$i > \$o ).

thf(exu_type,type,
exu: ( \$i > \$o ) > \$o ).

thf(exu,definition,
( exu
= ( ^ [Xphi: \$i > \$o] :
? [Xx: \$i] :
( ( Xphi @ Xx )
& ! [Xy: \$i] :
( ( Xphi @ Xy )
=> ( Xx = Xy ) ) ) ) ) ).

thf(descr_type,type,
descr: ( \$i > \$o ) > \$i ).

thf(descrp_type,type,
descrp: \$o ).

thf(descrp,definition,
( descrp
= ( ! [Xphi: \$i > \$o] :
( ( exu
@ ^ [Xx: \$i] : ( Xphi @ Xx ) )
=> ( Xphi
@ ( descr
@ ^ [Xx: \$i] : ( Xphi @ Xx ) ) ) ) ) ) ).

thf(in__Cong_type,type,
in__Cong: \$o ).

thf(in__Cong,definition,
( in__Cong
= ( ! [A: \$i,B: \$i] :
( ( A = B )
=> ! [Xx: \$i,Xy: \$i] :
( ( Xx = Xy )
=> ( ( in @ Xx @ A )
<=> ( in @ Xy @ B ) ) ) ) ) ) ).

thf(image1Ex1_type,type,
image1Ex1: \$o ).

thf(image1Ex1,definition,
( image1Ex1
= ( ! [A: \$i,Xf: \$i > \$i] :
( exu
@ ^ [B: \$i] :
! [Xx: \$i] :
( ( in @ Xx @ B )
<=> ? [Xy: \$i] :
( ( in @ Xy @ A )
& ( Xx
= ( Xf @ Xy ) ) ) ) ) ) ) ).

thf(image1_type,type,
image1: \$i > ( \$i > \$i ) > \$i ).

thf(image1,definition,
( image1
= ( ^ [A: \$i,Xf: \$i > \$i] :
( descr
@ ^ [B: \$i] :
! [Xx: \$i] :
( ( in @ Xx @ B )
<=> ? [Xy: \$i] :
( ( in @ Xy @ A )
& ( Xx
= ( Xf @ Xy ) ) ) ) ) ) ) ).

thf(image1Equiv,conjecture,
( descrp
=> ( in__Cong
=> ( image1Ex1
=> ! [A: \$i,Xf: \$i > \$i,Xx: \$i] :
( ( in @ Xx
@ ( image1 @ A
@ ^ [Xy: \$i] : ( Xf @ Xy ) ) )
<=> ? [Xy: \$i] :
( ( in @ Xy @ A )
& ( Xx
= ( Xf @ Xy ) ) ) ) ) ) ) ).

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