Theorem eqfnfv2f 5660

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 5656 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 5656	. 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 2336	. . . . 5 |- F/_ x z
4	2, 3	nffv 5565	. . . 4 |- F/_ x ( F ` z )
5		eqfnfv2f.2	. . . . 5 |- F/_ x G
6	5, 3	nffv 5565	. . . 4 |- F/_ x ( G ` z )
7	4, 6	nfeq 2344	. . 3 |- F/ x ( F ` z ) = ( G ` z )
8		nfv 1539	. . 3 |- F/ z ( F ` x ) = ( G ` x )
9		fveq2 5555	. . . 4 |- ( z = x -> ( F ` z ) = ( F ` x ) )
10		fveq2 5555	. . . 4 |- ( z = x -> ( G ` z ) = ( G ` x ) )
11	9, 10	eqeq12d 2208	. . 3 |- ( z = x -> ( ( F ` z ) = ( G ` z ) <-> ( F ` x ) = ( G ` x ) ) )
12	7, 8, 11	cbvral 2722	. 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 1364   F/_wnfc 2323   A.wral 2472   Fn wfn 5250   ` cfv 5255

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263

This theorem is referenced by: (None)