Metamath Proof Explorer

Theorem frr2

Description: Law of general well-founded recursion, part two. Now we establish the value of F within A . (Contributed by Scott Fenton, 11-Sep-2023)

Ref Expression
Hypothesis frr.1 
|- F = frecs ( R , A , G )
Assertion frr2 
|- ( ( ( R Fr A /\ R Se A ) /\ X e. A ) -> ( F ` X ) = ( X G ( F |` Pred ( R , A , X ) ) ) )

Proof

Step Hyp Ref Expression
1 frr.1 
 |-  F = frecs ( R , A , G )
2 1 frr1 
 |-  ( ( R Fr A /\ R Se A ) -> F Fn A )
3 2 fndmd 
 |-  ( ( R Fr A /\ R Se A ) -> dom F = A )
4 3 eleq2d 
 |-  ( ( R Fr A /\ R Se A ) -> ( X e. dom F <-> X e. A ) )
5 4 pm5.32i 
 |-  ( ( ( R Fr A /\ R Se A ) /\ X e. dom F ) <-> ( ( R Fr A /\ R Se A ) /\ X e. A ) )
6 fveq2 
 |-  ( y = X -> ( F ` y ) = ( F ` X ) )
7 id 
 |-  ( y = X -> y = X )
8 predeq3 
 |-  ( y = X -> Pred ( R , A , y ) = Pred ( R , A , X ) )
9 8 reseq2d 
 |-  ( y = X -> ( F |` Pred ( R , A , y ) ) = ( F |` Pred ( R , A , X ) ) )
10 7 9 oveq12d 
 |-  ( y = X -> ( y G ( F |` Pred ( R , A , y ) ) ) = ( X G ( F |` Pred ( R , A , X ) ) ) )
11 6 10 eqeq12d 
 |-  ( y = X -> ( ( F ` y ) = ( y G ( F |` Pred ( R , A , y ) ) ) <-> ( F ` X ) = ( X G ( F |` Pred ( R , A , X ) ) ) ) )
12 11 imbi2d 
 |-  ( y = X -> ( ( ( R Fr A /\ R Se A ) -> ( F ` y ) = ( y G ( F |` Pred ( R , A , y ) ) ) ) <-> ( ( R Fr A /\ R Se A ) -> ( F ` X ) = ( X G ( F |` Pred ( R , A , X ) ) ) ) ) )
13 eqid 
 |-  { a | E. b ( a Fn b /\ ( b C_ A /\ A. c e. b Pred ( R , A , c ) C_ b ) /\ A. c e. b ( a ` c ) = ( c G ( a |` Pred ( R , A , c ) ) ) ) } = { a | E. b ( a Fn b /\ ( b C_ A /\ A. c e. b Pred ( R , A , c ) C_ b ) /\ A. c e. b ( a ` c ) = ( c G ( a |` Pred ( R , A , c ) ) ) ) }
14 13 frrlem1 
 |-  { a | E. b ( a Fn b /\ ( b C_ A /\ A. c e. b Pred ( R , A , c ) C_ b ) /\ A. c e. b ( a ` c ) = ( c G ( a |` Pred ( R , A , c ) ) ) ) } = { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( y G ( f |` Pred ( R , A , y ) ) ) ) }
15 14 1 frrlem15 
 |-  ( ( ( R Fr A /\ R Se A ) /\ ( g e. { a | E. b ( a Fn b /\ ( b C_ A /\ A. c e. b Pred ( R , A , c ) C_ b ) /\ A. c e. b ( a ` c ) = ( c G ( a |` Pred ( R , A , c ) ) ) ) } /\ h e. { a | E. b ( a Fn b /\ ( b C_ A /\ A. c e. b Pred ( R , A , c ) C_ b ) /\ A. c e. b ( a ` c ) = ( c G ( a |` Pred ( R , A , c ) ) ) ) } ) ) -> ( ( x g u /\ x h v ) -> u = v ) )
16 14 1 15 frrlem10 
 |-  ( ( ( R Fr A /\ R Se A ) /\ y e. dom F ) -> ( F ` y ) = ( y G ( F |` Pred ( R , A , y ) ) ) )
17 16 expcom 
 |-  ( y e. dom F -> ( ( R Fr A /\ R Se A ) -> ( F ` y ) = ( y G ( F |` Pred ( R , A , y ) ) ) ) )
18 12 17 vtoclga 
 |-  ( X e. dom F -> ( ( R Fr A /\ R Se A ) -> ( F ` X ) = ( X G ( F |` Pred ( R , A , X ) ) ) ) )
19 18 impcom 
 |-  ( ( ( R Fr A /\ R Se A ) /\ X e. dom F ) -> ( F ` X ) = ( X G ( F |` Pred ( R , A , X ) ) ) )
20 5 19 sylbir 
 |-  ( ( ( R Fr A /\ R Se A ) /\ X e. A ) -> ( F ` X ) = ( X G ( F |` Pred ( R , A , X ) ) ) )