Theorem ffnfvf 5653

Description: A function maps to a class to which all values belong. This version of ffnfv 5652 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 5652	. 2 |- ( F : A --> B <-> ( F Fn A /\ A. z e. A ( F ` z ) e. B ) )
2		nfcv 2312	. . . 4 |- F/_ z A
3		ffnfvf.1	. . . 4 |- F/_ x A
4		ffnfvf.3	. . . . . 6 |- F/_ x F
5		nfcv 2312	. . . . . 6 |- F/_ x z
6	4, 5	nffv 5504	. . . . 5 |- F/_ x ( F ` z )
7		ffnfvf.2	. . . . 5 |- F/_ x B
8	6, 7	nfel 2321	. . . 4 |- F/ x ( F ` z ) e. B
9		nfv 1521	. . . 4 |- F/ z ( F ` x ) e. B
10		fveq2 5494	. . . . 5 |- ( z = x -> ( F ` z ) = ( F ` x ) )
11	10	eleq1d 2239	. . . 4 |- ( z = x -> ( ( F ` z ) e. B <-> ( F ` x ) e. B ) )
12	2, 3, 8, 9, 11	cbvralf 2689	. . 3 |- ( A. z e. A ( F ` z ) e. B <-> A. x e. A ( F ` x ) e. B )
13	12	anbi2i 454	. 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 2141   F/_wnfc 2299   A.wral 2448   Fn wfn 5191   -->wf 5192   ` cfv 5196

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204

This theorem is referenced by:  ixpf  6696  cc4f  7224