Theorem wfr3

Description: The principle of Well-Founded Recursion, part 3 of 3. Finally, we show that F is unique. We do this by showing that any function H with the same properties we proved of F in wfr1 and wfr2 is identical to F . (Contributed by Scott Fenton, 18-Apr-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	wfr3.1	|- R We A
	wfr3.2	|- R Se A
	wfr3.3	|- F = wrecs ( R , A , G )
Assertion	wfr3	|- ( ( H Fn A /\ A. z e. A ( H ` z ) = ( G ` ( H |` Pred ( R , A , z ) ) ) ) -> F = H )

Proof

Step	Hyp	Ref	Expression
1		wfr3.1	|- R We A
2		wfr3.2	|- R Se A
3		wfr3.3	|- F = wrecs ( R , A , G )
4	1 2	pm3.2i	|- ( R We A /\ R Se A )
5	1 2 3	wfr1	|- F Fn A
6	1 2 3	wfr2	|- ( z e. A -> ( F ` z ) = ( G ` ( F |` Pred ( R , A , z ) ) ) )
7	6	rgen	|- A. z e. A ( F ` z ) = ( G ` ( F |` Pred ( R , A , z ) ) )
8	5 7	pm3.2i	|- ( F Fn A /\ A. z e. A ( F ` z ) = ( G ` ( F |` Pred ( R , A , z ) ) ) )
9		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 )
10	4 8 9	mp3an12	|- ( ( H Fn A /\ A. z e. A ( H ` z ) = ( G ` ( H |` Pred ( R , A , z ) ) ) ) -> F = H )