Theorem fvreseq 5661

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 5367	. . . 4 |- ( ( F Fn A /\ B C_ A ) -> ( F |` B ) Fn B )
2		fnssres 5367	. . . 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 5655	. . 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 5578	. . . . 5 |- ( x e. B -> ( ( F |` B ) ` x ) = ( F ` x ) )
7		fvres 5578	. . . . 5 |- ( x e. B -> ( ( G |` B ) ` x ) = ( G ` x ) )
8	6, 7	eqeq12d 2208	. . . 4 |- ( x e. B -> ( ( ( F |` B ) ` x ) = ( ( G |` B ) ` x ) <-> ( F ` x ) = ( G ` x ) ) )
9	8	ralbiia 2508	. . 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 2164   A.wral 2472   C_ wss 3153   |` cres 4661   Fn wfn 5249   ` cfv 5254

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262

This theorem is referenced by:  tfri3  6420