Theorem eqfnfv2f 5704

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 5700 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 5700	. 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 2350	. . . . 5 |- F/_ x z
4	2, 3	nffv 5609	. . . 4 |- F/_ x ( F ` z )
5		eqfnfv2f.2	. . . . 5 |- F/_ x G
6	5, 3	nffv 5609	. . . 4 |- F/_ x ( G ` z )
7	4, 6	nfeq 2358	. . 3 |- F/ x ( F ` z ) = ( G ` z )
8		nfv 1552	. . 3 |- F/ z ( F ` x ) = ( G ` x )
9		fveq2 5599	. . . 4 |- ( z = x -> ( F ` z ) = ( F ` x ) )
10		fveq2 5599	. . . 4 |- ( z = x -> ( G ` z ) = ( G ` x ) )
11	9, 10	eqeq12d 2222	. . 3 |- ( z = x -> ( ( F ` z ) = ( G ` z ) <-> ( F ` x ) = ( G ` x ) ) )
12	7, 8, 11	cbvral 2738	. 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 1373   F/_wnfc 2337   A.wral 2486   Fn wfn 5285   ` cfv 5290

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298

This theorem is referenced by: (None)