Theorem fvreseq 5668

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 5374	. . . 4 |- ( ( F Fn A /\ B C_ A ) -> ( F |` B ) Fn B )
2		fnssres 5374	. . . 4 |- ( ( G Fn A /\ B C_ A ) -> ( G |` B ) Fn B )
3	1, 2	anim12i 338	. . 3 |- ( ( ( F Fn A /\ B C_ A ) /\ ( G Fn A /\ B C_ A ) ) -> ( ( F |` B ) Fn B /\ ( G |` B ) Fn B ) )
4	3	anandirs 593	. 2 |- ( ( ( F Fn A /\ G Fn A ) /\ B C_ A ) -> ( ( F |` B ) Fn B /\ ( G |` B ) Fn B ) )
5		eqfnfv 5662	. . 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 5585	. . . . 5 |- ( x e. B -> ( ( F |` B ) ` x ) = ( F ` x ) )
7		fvres 5585	. . . . 5 |- ( x e. B -> ( ( G |` B ) ` x ) = ( G ` x ) )
8	6, 7	eqeq12d 2211	. . . 4 |- ( x e. B -> ( ( ( F |` B ) ` x ) = ( ( G |` B ) ` x ) <-> ( F ` x ) = ( G ` x ) ) )
9	8	ralbiia 2511	. . 3 |- ( A. x e. B ( ( F |` B ) ` x ) = ( ( G |` B ) ` x ) <-> A. x e. B ( F ` x ) = ( G ` x ) )
10	5, 9	bitrdi 196	. 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 104   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   C_ wss 3157   |` cres 4666   Fn wfn 5254   ` cfv 5259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267

This theorem is referenced by:  tfri3  6434