Theorem eqfnfv3 5661

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 5660	. 2 |- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) )
2		eqss 3198	. . . . 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 3173	. . . . . . 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 2623	. . . . . 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 1364   e. wcel 2167   A.wral 2475   C_ wss 3157   Fn wfn 5253   ` cfv 5258

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266

This theorem is referenced by: (None)