Theorem eqfnfv2f 5597

Description: Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5593 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)

Hypotheses

Ref	Expression
eqfnfv2f.1	|- F/_ x F
eqfnfv2f.2	|- F/_ x G

Assertion

Ref	Expression
eqfnfv2f	|- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )

Distinct variable group:   x, A

Allowed substitution hints:   F( x)   G( x)

Proof of Theorem eqfnfv2f

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqfnfv 5593	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. z e. A ( F ` z ) = ( G ` z ) ) )
2		eqfnfv2f.1	. . . . 5 |- F/_ x F
3		nfcv 2312	. . . . 5 |- F/_ x z
4	2, 3	nffv 5506	. . . 4 |- F/_ x ( F ` z )
5		eqfnfv2f.2	. . . . 5 |- F/_ x G
6	5, 3	nffv 5506	. . . 4 |- F/_ x ( G ` z )
7	4, 6	nfeq 2320	. . 3 |- F/ x ( F ` z ) = ( G ` z )
8		nfv 1521	. . 3 |- F/ z ( F ` x ) = ( G ` x )
9		fveq2 5496	. . . 4 |- ( z = x -> ( F ` z ) = ( F ` x ) )
10		fveq2 5496	. . . 4 |- ( z = x -> ( G ` z ) = ( G ` x ) )
11	9, 10	eqeq12d 2185	. . 3 |- ( z = x -> ( ( F ` z ) = ( G ` z ) <-> ( F ` x ) = ( G ` x ) ) )
12	7, 8, 11	cbvral 2692	. 2 |- ( A. z e. A ( F ` z ) = ( G ` z ) <-> A. x e. A ( F ` x ) = ( G ` x ) )
13	1, 12	bitrdi 195	1 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   F/_wnfc 2299   A.wral 2448   Fn wfn 5193   ` cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by: (None)