Theorem wrecseq123 7408

Description: General equality theorem for the well-founded recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.)

Assertion

Ref	Expression
wrecseq123	|- ( ( R = S /\ A = B /\ F = G ) -> wrecs ( R , A , F ) = wrecs ( S , B , G ) )

Proof of Theorem wrecseq123

Dummy variables f x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 3627	. . . . . . . 8 |- ( A = B -> ( x C_ A <-> x C_ B ) )
2	1	3ad2ant2 1083	. . . . . . 7 |- ( ( R = S /\ A = B /\ F = G ) -> ( x C_ A <-> x C_ B ) )
3		predeq1 5682	. . . . . . . . . . 11 |- ( R = S -> Pred ( R , A , y ) = Pred ( S , A , y ) )
4		predeq2 5683	. . . . . . . . . . 11 |- ( A = B -> Pred ( S , A , y ) = Pred ( S , B , y ) )
5	3, 4	sylan9eq 2676	. . . . . . . . . 10 |- ( ( R = S /\ A = B ) -> Pred ( R , A , y ) = Pred ( S , B , y ) )
6	5	3adant3 1081	. . . . . . . . 9 |- ( ( R = S /\ A = B /\ F = G ) -> Pred ( R , A , y ) = Pred ( S , B , y ) )
7	6	sseq1d 3632	. . . . . . . 8 |- ( ( R = S /\ A = B /\ F = G ) -> ( Pred ( R , A , y ) C_ x <-> Pred ( S , B , y ) C_ x ) )
8	7	ralbidv 2986	. . . . . . 7 |- ( ( R = S /\ A = B /\ F = G ) -> ( A. y e. x Pred ( R , A , y ) C_ x <-> A. y e. x Pred ( S , B , y ) C_ x ) )
9	2, 8	anbi12d 747	. . . . . 6 |- ( ( R = S /\ A = B /\ F = G ) -> ( ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) <-> ( x C_ B /\ A. y e. x Pred ( S , B , y ) C_ x ) ) )
10		simp3 1063	. . . . . . . . 9 |- ( ( R = S /\ A = B /\ F = G ) -> F = G )
11	5	reseq2d 5396	. . . . . . . . . 10 |- ( ( R = S /\ A = B ) -> ( f |` Pred ( R , A , y ) ) = ( f |` Pred ( S , B , y ) ) )
12	11	3adant3 1081	. . . . . . . . 9 |- ( ( R = S /\ A = B /\ F = G ) -> ( f |` Pred ( R , A , y ) ) = ( f |` Pred ( S , B , y ) ) )
13	10, 12	fveq12d 6197	. . . . . . . 8 |- ( ( R = S /\ A = B /\ F = G ) -> ( F ` ( f |` Pred ( R , A , y ) ) ) = ( G ` ( f |` Pred ( S , B , y ) ) ) )
14	13	eqeq2d 2632	. . . . . . 7 |- ( ( R = S /\ A = B /\ F = G ) -> ( ( f ` y ) = ( F ` ( f |` Pred ( R , A , y ) ) ) <-> ( f ` y ) = ( G ` ( f |` Pred ( S , B , y ) ) ) ) )
15	14	ralbidv 2986	. . . . . 6 |- ( ( R = S /\ A = B /\ F = G ) -> ( A. y e. x ( f ` y ) = ( F ` ( f |` Pred ( R , A , y ) ) ) <-> A. y e. x ( f ` y ) = ( G ` ( f |` Pred ( S , B , y ) ) ) ) )
16	9, 15	3anbi23d 1402	. . . . 5 |- ( ( R = S /\ A = B /\ F = G ) -> ( ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( F ` ( f |` Pred ( R , A , y ) ) ) ) <-> ( f Fn x /\ ( x C_ B /\ A. y e. x Pred ( S , B , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( G ` ( f |` Pred ( S , B , y ) ) ) ) ) )
17	16	exbidv 1850	. . . 4 |- ( ( R = S /\ A = B /\ F = G ) -> ( E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( F ` ( f |` Pred ( R , A , y ) ) ) ) <-> E. x ( f Fn x /\ ( x C_ B /\ A. y e. x Pred ( S , B , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( G ` ( f |` Pred ( S , B , y ) ) ) ) ) )
18	17	abbidv 2741	. . 3 |- ( ( R = S /\ A = B /\ F = G ) -> { 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 ) = ( F ` ( f |` Pred ( R , A , y ) ) ) ) } = { f | E. x ( f Fn x /\ ( x C_ B /\ A. y e. x Pred ( S , B , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( G ` ( f |` Pred ( S , B , y ) ) ) ) } )
19	18	unieqd 4446	. 2 |- ( ( R = S /\ A = B /\ F = G ) -> U. { 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 ) = ( F ` ( f |` Pred ( R , A , y ) ) ) ) } = U. { f | E. x ( f Fn x /\ ( x C_ B /\ A. y e. x Pred ( S , B , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( G ` ( f |` Pred ( S , B , y ) ) ) ) } )
20		df-wrecs 7407	. 2 |- wrecs ( R , A , F ) = U. { 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 ) = ( F ` ( f |` Pred ( R , A , y ) ) ) ) }
21		df-wrecs 7407	. 2 |- wrecs ( S , B , G ) = U. { f | E. x ( f Fn x /\ ( x C_ B /\ A. y e. x Pred ( S , B , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( G ` ( f |` Pred ( S , B , y ) ) ) ) }
22	19, 20, 21	3eqtr4g 2681	1 |- ( ( R = S /\ A = B /\ F = G ) -> wrecs ( R , A , F ) = wrecs ( S , B , G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   {cab 2608   A.wral 2912   C_ wss 3574   U.cuni 4436   |` cres 5116   Predcpred 5679   Fn wfn 5883   ` cfv 5888  wrecscwrecs 7406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fv 5896  df-wrecs 7407

This theorem is referenced by:  wrecseq1  7410  wrecseq2  7411  wrecseq3  7412