Theorem eqfnfv3 5749

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 5748	. 2 |- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) )
2		eqss 3241	. . . . 5 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
3		ancom 266	. . . . 5 |- ( ( A C_ B /\ B C_ A ) <-> ( B C_ A /\ A C_ B ) )
4	2, 3	bitri 184	. . . 4 |- ( A = B <-> ( B C_ A /\ A C_ B ) )
5	4	anbi1i 458	. . 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 401	. . . 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 3215	. . . . . . 7 |- ( A C_ B <-> A. x e. A x e. B )
8	7	anbi1i 458	. . . . . 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 2658	. . . . . 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 187	. . . . 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 457	. . . 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 184	. . 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 184	. 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	bitrdi 196	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 104   <-> wb 105   = wceq 1397   e. wcel 2201   A.wral 2509   C_ wss 3199   Fn wfn 5323   ` cfv 5328

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-csb 3127  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-iota 5288  df-fun 5330  df-fn 5331  df-fv 5336

This theorem is referenced by: (None)