Theorem eqfnfv3 5734

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 5733	. 2 |- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) )
2		eqss 3239	. . . . 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 3213	. . . . . . 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 2657	. . . . . 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 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   Fn wfn 5313   ` cfv 5318

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 4202  ax-pow 4258  ax-pr 4293

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 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326

This theorem is referenced by: (None)