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Theorem ffnfvf 5827
Description: A function maps to a class to which all values belong. This version of ffnfv 5826 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.
StepHypRef Expression
1 ffnfv 5826 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. z  e.  A  ( F `  z )  e.  B
) )
2 nfcv 2516 . . . 4  |-  F/_ z A
3 ffnfvf.1 . . . 4  |-  F/_ x A
4 ffnfvf.3 . . . . . 6  |-  F/_ x F
5 nfcv 2516 . . . . . 6  |-  F/_ x
z
64, 5nffv 5668 . . . . 5  |-  F/_ x
( F `  z
)
7 ffnfvf.2 . . . . 5  |-  F/_ x B
86, 7nfel 2524 . . . 4  |-  F/ x
( F `  z
)  e.  B
9 nfv 1626 . . . 4  |-  F/ z ( F `  x
)  e.  B
10 fveq2 5661 . . . . 5  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
1110eleq1d 2446 . . . 4  |-  ( z  =  x  ->  (
( F `  z
)  e.  B  <->  ( F `  x )  e.  B
) )
122, 3, 8, 9, 11cbvralf 2862 . . 3  |-  ( A. z  e.  A  ( F `  z )  e.  B  <->  A. x  e.  A  ( F `  x )  e.  B )
1312anbi2i 676 . 2  |-  ( ( F  Fn  A  /\  A. z  e.  A  ( F `  z )  e.  B )  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) )
141, 13bitri 241 1  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1717   F/_wnfc 2503   A.wral 2642    Fn wfn 5382   -->wf 5383   ` cfv 5387
This theorem is referenced by:  ixpf  7013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395
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