Theorem fvreseq 5599

Description: Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)

Assertion

Ref	Expression
fvreseq	|- ( ( ( F Fn A /\ G Fn A ) /\ B C_ A ) -> ( ( F |` B ) = ( G |` B ) <-> A. x e. B ( F ` x ) = ( G ` x ) ) )

Distinct variable groups:   x, B   x, F   x, G

Allowed substitution hint:   A( x)

Proof of Theorem fvreseq

Step	Hyp	Ref	Expression
1		fnssres 5311	. . . 4 |- ( ( F Fn A /\ B C_ A ) -> ( F |` B ) Fn B )
2		fnssres 5311	. . . 4 |- ( ( G Fn A /\ B C_ A ) -> ( G |` B ) Fn B )
3	1, 2	anim12i 336	. . 3 |- ( ( ( F Fn A /\ B C_ A ) /\ ( G Fn A /\ B C_ A ) ) -> ( ( F |` B ) Fn B /\ ( G |` B ) Fn B ) )
4	3	anandirs 588	. 2 |- ( ( ( F Fn A /\ G Fn A ) /\ B C_ A ) -> ( ( F |` B ) Fn B /\ ( G |` B ) Fn B ) )
5		eqfnfv 5593	. . 3 |- ( ( ( F |` B ) Fn B /\ ( G |` B ) Fn B ) -> ( ( F |` B ) = ( G |` B ) <-> A. x e. B ( ( F |` B ) ` x ) = ( ( G |` B ) ` x ) ) )
6		fvres 5520	. . . . 5 |- ( x e. B -> ( ( F |` B ) ` x ) = ( F ` x ) )
7		fvres 5520	. . . . 5 |- ( x e. B -> ( ( G |` B ) ` x ) = ( G ` x ) )
8	6, 7	eqeq12d 2185	. . . 4 |- ( x e. B -> ( ( ( F |` B ) ` x ) = ( ( G |` B ) ` x ) <-> ( F ` x ) = ( G ` x ) ) )
9	8	ralbiia 2484	. . 3 |- ( A. x e. B ( ( F |` B ) ` x ) = ( ( G |` B ) ` x ) <-> A. x e. B ( F ` x ) = ( G ` x ) )
10	5, 9	bitrdi 195	. 2 |- ( ( ( F |` B ) Fn B /\ ( G |` B ) Fn B ) -> ( ( F |` B ) = ( G |` B ) <-> A. x e. B ( F ` x ) = ( G ` x ) ) )
11	4, 10	syl 14	1 |- ( ( ( F Fn A /\ G Fn A ) /\ B C_ A ) -> ( ( F |` B ) = ( G |` B ) <-> A. x e. B ( F ` x ) = ( G ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   C_ wss 3121   |` cres 4613   Fn wfn 5193   ` cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by:  tfri3  6346