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 recursively applied to all "previous" values of F . (Contributed by Scott Fenton, 18-Apr-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	wfr2.1	|- R We A
	wfr2.2	|- R Se A
	wfr2.3	|- F = wrecs ( R , A , G )
Assertion	wfr2	|- ( X e. A -> ( F ` X ) = ( G ` ( F |` Pred ( R , A , X ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		wfr2.1	|- R We A
2		wfr2.2	|- R Se A
3		wfr2.3	|- F = wrecs ( R , A , G )
4		eqid	|- ( F u. { <. x , ( G ` ( F |` Pred ( R , A , x ) ) ) >. } ) = ( F u. { <. x , ( G ` ( F |` Pred ( R , A , x ) ) ) >. } )
5	1 2 3 4	wfrlem16	|- dom F = A
6	5	eleq2i	|- ( X e. dom F <-> X e. A )
7	1 2 3	wfr2a	|- ( X e. dom F -> ( F ` X ) = ( G ` ( F |` Pred ( R , A , X ) ) ) )
8	6 7	sylbir	|- ( X e. A -> ( F ` X ) = ( G ` ( F |` Pred ( R , A , X ) ) ) )