Theorem ffnfvf 5721

Description: A function maps to a class to which all values belong. This version of ffnfv 5720 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)

Hypotheses

Ref	Expression
ffnfvf.1	|- F/_ x A
ffnfvf.2	|- F/_ x B
ffnfvf.3	|- F/_ x F

Assertion

Ref	Expression
ffnfvf	|- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )

Proof of Theorem ffnfvf

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffnfv 5720	. 2 |- ( F : A --> B <-> ( F Fn A /\ A. z e. A ( F ` z ) e. B ) )
2		nfcv 2339	. . . 4 |- F/_ z A
3		ffnfvf.1	. . . 4 |- F/_ x A
4		ffnfvf.3	. . . . . 6 |- F/_ x F
5		nfcv 2339	. . . . . 6 |- F/_ x z
6	4, 5	nffv 5568	. . . . 5 |- F/_ x ( F ` z )
7		ffnfvf.2	. . . . 5 |- F/_ x B
8	6, 7	nfel 2348	. . . 4 |- F/ x ( F ` z ) e. B
9		nfv 1542	. . . 4 |- F/ z ( F ` x ) e. B
10		fveq2 5558	. . . . 5 |- ( z = x -> ( F ` z ) = ( F ` x ) )
11	10	eleq1d 2265	. . . 4 |- ( z = x -> ( ( F ` z ) e. B <-> ( F ` x ) e. B ) )
12	2, 3, 8, 9, 11	cbvralf 2721	. . 3 |- ( A. z e. A ( F ` z ) e. B <-> A. x e. A ( F ` x ) e. B )
13	12	anbi2i 457	. 2 |- ( ( F Fn A /\ A. z e. A ( F ` z ) e. B ) <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )
14	1, 13	bitri 184	1 |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2167   F/_wnfc 2326   A.wral 2475   Fn wfn 5253   -->wf 5254   ` cfv 5258

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266

This theorem is referenced by:  ixpf  6779  cc4f  7336