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Theorem ffnfvf 5887
Description: A function maps to a class to which all values belong. This version of ffnfv 5886 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 5886 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. z  e.  A  ( F `  z )  e.  B
) )
2 nfcv 2571 . . . 4  |-  F/_ z A
3 ffnfvf.1 . . . 4  |-  F/_ x A
4 ffnfvf.3 . . . . . 6  |-  F/_ x F
5 nfcv 2571 . . . . . 6  |-  F/_ x
z
64, 5nffv 5727 . . . . 5  |-  F/_ x
( F `  z
)
7 ffnfvf.2 . . . . 5  |-  F/_ x B
86, 7nfel 2579 . . . 4  |-  F/ x
( F `  z
)  e.  B
9 nfv 1629 . . . 4  |-  F/ z ( F `  x
)  e.  B
10 fveq2 5720 . . . . 5  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
1110eleq1d 2501 . . . 4  |-  ( z  =  x  ->  (
( F `  z
)  e.  B  <->  ( F `  x )  e.  B
) )
122, 3, 8, 9, 11cbvralf 2918 . . 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 1725   F/_wnfc 2558   A.wral 2697    Fn wfn 5441   -->wf 5442   ` cfv 5446
This theorem is referenced by:  ixpf  7076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454
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