Theorem eqfnfv2f 5290

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 5286 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 5286	. 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 2219	. . . . 5 |- F/_ x z
4	2, 3	nffv 5205	. . . 4 |- F/_ x ( F ` z )
5		eqfnfv2f.2	. . . . 5 |- F/_ x G
6	5, 3	nffv 5205	. . . 4 |- F/_ x ( G ` z )
7	4, 6	nfeq 2226	. . 3 |- F/ x ( F ` z ) = ( G ` z )
8		nfv 1461	. . 3 |- F/ z ( F ` x ) = ( G ` x )
9		fveq2 5198	. . . 4 |- ( z = x -> ( F ` z ) = ( F ` x ) )
10		fveq2 5198	. . . 4 |- ( z = x -> ( G ` z ) = ( G ` x ) )
11	9, 10	eqeq12d 2095	. . 3 |- ( z = x -> ( ( F ` z ) = ( G ` z ) <-> ( F ` x ) = ( G ` x ) ) )
12	7, 8, 11	cbvral 2573	. 2 |- ( A. z e. A ( F ` z ) = ( G ` z ) <-> A. x e. A ( F ` x ) = ( G ` x ) )
13	1, 12	syl6bb 194	1 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   F/_wnfc 2206   A.wral 2348   Fn wfn 4917   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930

This theorem is referenced by: (None)