Theorem fvreseq 5746

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 5442	. . . 4 |- ( ( F Fn A /\ B C_ A ) -> ( F |` B ) Fn B )
2		fnssres 5442	. . . 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 595	. 2 |- ( ( ( F Fn A /\ G Fn A ) /\ B C_ A ) -> ( ( F |` B ) Fn B /\ ( G |` B ) Fn B ) )
5		eqfnfv 5740	. . 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 5659	. . . . 5 |- ( x e. B -> ( ( F |` B ) ` x ) = ( F ` x ) )
7		fvres 5659	. . . . 5 |- ( x e. B -> ( ( G |` B ) ` x ) = ( G ` x ) )
8	6, 7	eqeq12d 2244	. . . 4 |- ( x e. B -> ( ( ( F |` B ) ` x ) = ( ( G |` B ) ` x ) <-> ( F ` x ) = ( G ` x ) ) )
9	8	ralbiia 2544	. . 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 1395   e. wcel 2200   A.wral 2508   C_ wss 3198   |` cres 4725   Fn wfn 5319   ` cfv 5324

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332

This theorem is referenced by:  tfri3  6528