Theorem eqfnfv2f 5779

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 5775 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 5775	. 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 2384	. . . . 5 |- F/_ x z
4	2, 3	nffv 5680	. . . 4 |- F/_ x ( F ` z )
5		eqfnfv2f.2	. . . . 5 |- F/_ x G
6	5, 3	nffv 5680	. . . 4 |- F/_ x ( G ` z )
7	4, 6	nfeq 2392	. . 3 |- F/ x ( F ` z ) = ( G ` z )
8		nfv 1577	. . 3 |- F/ z ( F ` x ) = ( G ` x )
9		fveq2 5670	. . . 4 |- ( z = x -> ( F ` z ) = ( F ` x ) )
10		fveq2 5670	. . . 4 |- ( z = x -> ( G ` z ) = ( G ` x ) )
11	9, 10	eqeq12d 2247	. . 3 |- ( z = x -> ( ( F ` z ) = ( G ` z ) <-> ( F ` x ) = ( G ` x ) ) )
12	7, 8, 11	cbvral 2774	. 2 |- ( A. z e. A ( F ` z ) = ( G ` z ) <-> A. x e. A ( F ` x ) = ( G ` x ) )
13	1, 12	bitrdi 196	1 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   F/_wnfc 2371   A.wral 2520   Fn wfn 5347   ` cfv 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360

This theorem is referenced by: (None)