Theorem ffnfvf 5794

Description: A function maps to a class to which all values belong. This version of ffnfv 5793 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 5793	. 2 |- ( F : A --> B <-> ( F Fn A /\ A. z e. A ( F ` z ) e. B ) )
2		nfcv 2372	. . . 4 |- F/_ z A
3		ffnfvf.1	. . . 4 |- F/_ x A
4		ffnfvf.3	. . . . . 6 |- F/_ x F
5		nfcv 2372	. . . . . 6 |- F/_ x z
6	4, 5	nffv 5637	. . . . 5 |- F/_ x ( F ` z )
7		ffnfvf.2	. . . . 5 |- F/_ x B
8	6, 7	nfel 2381	. . . 4 |- F/ x ( F ` z ) e. B
9		nfv 1574	. . . 4 |- F/ z ( F ` x ) e. B
10		fveq2 5627	. . . . 5 |- ( z = x -> ( F ` z ) = ( F ` x ) )
11	10	eleq1d 2298	. . . 4 |- ( z = x -> ( ( F ` z ) e. B <-> ( F ` x ) e. B ) )
12	2, 3, 8, 9, 11	cbvralf 2756	. . 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 2200   F/_wnfc 2359   A.wral 2508   Fn wfn 5313   -->wf 5314   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326

This theorem is referenced by:  ixpf  6867  cc4f  7455