Theorem eqfnfv3 5267

Description: Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
eqfnfv3	|- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( B C_ A /\ A. x e. A ( x e. B /\ ( F ` x ) = ( G ` x ) ) ) ) )

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

Proof of Theorem eqfnfv3

Step	Hyp	Ref	Expression
1		eqfnfv2 5266	. 2 |- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) )
2		eqss 2960	. . . . 5 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
3		ancom 253	. . . . 5 |- ( ( A C_ B /\ B C_ A ) <-> ( B C_ A /\ A C_ B ) )
4	2, 3	bitri 173	. . . 4 |- ( A = B <-> ( B C_ A /\ A C_ B ) )
5	4	anbi1i 431	. . 3 |- ( ( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) <-> ( ( B C_ A /\ A C_ B ) /\ A. x e. A ( F ` x ) = ( G ` x ) ) )
6		anass 381	. . . 4 |- ( ( ( B C_ A /\ A C_ B ) /\ A. x e. A ( F ` x ) = ( G ` x ) ) <-> ( B C_ A /\ ( A C_ B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) )
7		dfss3 2935	. . . . . . 7 |- ( A C_ B <-> A. x e. A x e. B )
8	7	anbi1i 431	. . . . . 6 |- ( ( A C_ B /\ A. x e. A ( F ` x ) = ( G ` x ) ) <-> ( A. x e. A x e. B /\ A. x e. A ( F ` x ) = ( G ` x ) ) )
9		r19.26 2441	. . . . . 6 |- ( A. x e. A ( x e. B /\ ( F ` x ) = ( G ` x ) ) <-> ( A. x e. A x e. B /\ A. x e. A ( F ` x ) = ( G ` x ) ) )
10	8, 9	bitr4i 176	. . . . 5 |- ( ( A C_ B /\ A. x e. A ( F ` x ) = ( G ` x ) ) <-> A. x e. A ( x e. B /\ ( F ` x ) = ( G ` x ) ) )
11	10	anbi2i 430	. . . 4 |- ( ( B C_ A /\ ( A C_ B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) <-> ( B C_ A /\ A. x e. A ( x e. B /\ ( F ` x ) = ( G ` x ) ) ) )
12	6, 11	bitri 173	. . 3 |- ( ( ( B C_ A /\ A C_ B ) /\ A. x e. A ( F ` x ) = ( G ` x ) ) <-> ( B C_ A /\ A. x e. A ( x e. B /\ ( F ` x ) = ( G ` x ) ) ) )
13	5, 12	bitri 173	. 2 |- ( ( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) <-> ( B C_ A /\ A. x e. A ( x e. B /\ ( F ` x ) = ( G ` x ) ) ) )
14	1, 13	syl6bb 185	1 |- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( B C_ A /\ A. x e. A ( x e. B /\ ( F ` x ) = ( G ` x ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   A.wral 2306   C_ wss 2917   Fn wfn 4897   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910

This theorem is referenced by: (None)