Theorem ffnfvf 5579

Description: A function maps to a class to which all values belong. This version of ffnfv 5578 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 5578	. 2 |- ( F : A --> B <-> ( F Fn A /\ A. z e. A ( F ` z ) e. B ) )
2		nfcv 2281	. . . 4 |- F/_ z A
3		ffnfvf.1	. . . 4 |- F/_ x A
4		ffnfvf.3	. . . . . 6 |- F/_ x F
5		nfcv 2281	. . . . . 6 |- F/_ x z
6	4, 5	nffv 5431	. . . . 5 |- F/_ x ( F ` z )
7		ffnfvf.2	. . . . 5 |- F/_ x B
8	6, 7	nfel 2290	. . . 4 |- F/ x ( F ` z ) e. B
9		nfv 1508	. . . 4 |- F/ z ( F ` x ) e. B
10		fveq2 5421	. . . . 5 |- ( z = x -> ( F ` z ) = ( F ` x ) )
11	10	eleq1d 2208	. . . 4 |- ( z = x -> ( ( F ` z ) e. B <-> ( F ` x ) e. B ) )
12	2, 3, 8, 9, 11	cbvralf 2648	. . 3 |- ( A. z e. A ( F ` z ) e. B <-> A. x e. A ( F ` x ) e. B )
13	12	anbi2i 452	. 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 183	1 |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )

Syntax hints:   /\ wa 103   <-> wb 104   e. wcel 1480   F/_wnfc 2268   A.wral 2416   Fn wfn 5118   -->wf 5119   ` cfv 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131

This theorem is referenced by:  ixpf  6614