Theorem wfr2

Description: The Principle of Well-Ordered Recursion, part 2 of 3. Next, we show that the value of F at any X e. A is G applied to all "previous" values of F . (Contributed by Scott Fenton, 18-Apr-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypothesis	wfr2.1	|- F = wrecs ( R , A , G )
Assertion	wfr2	|- ( ( ( R We A /\ R Se A ) /\ X e. A ) -> ( F ` X ) = ( G ` ( F |` Pred ( R , A , X ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		wfr2.1	|- F = wrecs ( R , A , G )
2	1	wfr1	|- ( ( R We A /\ R Se A ) -> F Fn A )
3	2	fndmd	|- ( ( R We A /\ R Se A ) -> dom F = A )
4	3	eleq2d	|- ( ( R We A /\ R Se A ) -> ( X e. dom F <-> X e. A ) )
5	4	biimpar	|- ( ( ( R We A /\ R Se A ) /\ X e. A ) -> X e. dom F )
6	1	wfr2a	|- ( ( ( R We A /\ R Se A ) /\ X e. dom F ) -> ( F ` X ) = ( G ` ( F |` Pred ( R , A , X ) ) ) )
7	5 6	syldan	|- ( ( ( R We A /\ R Se A ) /\ X e. A ) -> ( F ` X ) = ( G ` ( F |` Pred ( R , A , X ) ) ) )