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Theorem ffnfvf 5686
Description: A function maps to a class to which all values belong. This version of ffnfv 5685 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 5685 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. z  e.  A  ( F `  z )  e.  B
) )
2 nfcv 2419 . . . 4  |-  F/_ z A
3 ffnfvf.1 . . . 4  |-  F/_ x A
4 ffnfvf.3 . . . . . 6  |-  F/_ x F
5 nfcv 2419 . . . . . 6  |-  F/_ x
z
64, 5nffv 5532 . . . . 5  |-  F/_ x
( F `  z
)
7 ffnfvf.2 . . . . 5  |-  F/_ x B
86, 7nfel 2427 . . . 4  |-  F/ x
( F `  z
)  e.  B
9 nfv 1605 . . . 4  |-  F/ z ( F `  x
)  e.  B
10 fveq2 5525 . . . . 5  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
1110eleq1d 2349 . . . 4  |-  ( z  =  x  ->  (
( F `  z
)  e.  B  <->  ( F `  x )  e.  B
) )
122, 3, 8, 9, 11cbvralf 2758 . . 3  |-  ( A. z  e.  A  ( F `  z )  e.  B  <->  A. x  e.  A  ( F `  x )  e.  B )
1312anbi2i 675 . 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 240 1  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   F/_wnfc 2406   A.wral 2543    Fn wfn 5250   -->wf 5251   ` cfv 5255
This theorem is referenced by:  ixpf  6838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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