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Metamath Proof Explorer

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 ) ) ) )