Theorem ffnfvf 5677

Description: A function maps to a class to which all values belong. This version of ffnfv 5676 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 5676	. 2 |- ( F : A --> B <-> ( F Fn A /\ A. z e. A ( F ` z ) e. B ) )
2		nfcv 2319	. . . 4 |- F/_ z A
3		ffnfvf.1	. . . 4 |- F/_ x A
4		ffnfvf.3	. . . . . 6 |- F/_ x F
5		nfcv 2319	. . . . . 6 |- F/_ x z
6	4, 5	nffv 5527	. . . . 5 |- F/_ x ( F ` z )
7		ffnfvf.2	. . . . 5 |- F/_ x B
8	6, 7	nfel 2328	. . . 4 |- F/ x ( F ` z ) e. B
9		nfv 1528	. . . 4 |- F/ z ( F ` x ) e. B
10		fveq2 5517	. . . . 5 |- ( z = x -> ( F ` z ) = ( F ` x ) )
11	10	eleq1d 2246	. . . 4 |- ( z = x -> ( ( F ` z ) e. B <-> ( F ` x ) e. B ) )
12	2, 3, 8, 9, 11	cbvralf 2697	. . 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 2148   F/_wnfc 2306   A.wral 2455   Fn wfn 5213   -->wf 5214   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226

This theorem is referenced by:  ixpf  6722  cc4f  7270