Theorem wfr3OLD

Description: Obsolete form of wfr3 as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 18-Apr-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	wfr3OLD.1	|- R We A
	wfr3OLD.2	|- R Se A
	wfr3OLD.3	|- F = wrecs ( R , A , G )
Assertion	wfr3OLD	|- ( ( H Fn A /\ A. z e. A ( H ` z ) = ( G ` ( H |` Pred ( R , A , z ) ) ) ) -> F = H )

Proof

Step	Hyp	Ref	Expression
1		wfr3OLD.1	|- R We A
2		wfr3OLD.2	|- R Se A
3		wfr3OLD.3	|- F = wrecs ( R , A , G )
4	1 2	pm3.2i	|- ( R We A /\ R Se A )
5	3	wfr1	|- ( ( R We A /\ R Se A ) -> F Fn A )
6	1 2 5	mp2an	|- F Fn A
7	3	wfr2	|- ( ( ( R We A /\ R Se A ) /\ z e. A ) -> ( F ` z ) = ( G ` ( F |` Pred ( R , A , z ) ) ) )
8	1 2 7	mpanl12	|- ( z e. A -> ( F ` z ) = ( G ` ( F |` Pred ( R , A , z ) ) ) )
9	8	rgen	|- A. z e. A ( F ` z ) = ( G ` ( F |` Pred ( R , A , z ) ) )
10	6 9	pm3.2i	|- ( F Fn A /\ A. z e. A ( F ` z ) = ( G ` ( F |` Pred ( R , A , z ) ) ) )
11		wfr3g	|- ( ( ( R We A /\ R Se A ) /\ ( F Fn A /\ A. z e. A ( F ` z ) = ( G ` ( F |` Pred ( R , A , z ) ) ) ) /\ ( H Fn A /\ A. z e. A ( H ` z ) = ( G ` ( H |` Pred ( R , A , z ) ) ) ) ) -> F = H )
12	4 10 11	mp3an12	|- ( ( H Fn A /\ A. z e. A ( H ` z ) = ( G ` ( H |` Pred ( R , A , z ) ) ) ) -> F = H )