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Theorem ffnfvf 5696
Description: A function maps to a class to which all values belong. This version of ffnfv 5695 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 5695 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. z  e.  A  ( F `  z )  e.  B
) )
2 nfcv 2332 . . . 4  |-  F/_ z A
3 ffnfvf.1 . . . 4  |-  F/_ x A
4 ffnfvf.3 . . . . . 6  |-  F/_ x F
5 nfcv 2332 . . . . . 6  |-  F/_ x
z
64, 5nffv 5544 . . . . 5  |-  F/_ x
( F `  z
)
7 ffnfvf.2 . . . . 5  |-  F/_ x B
86, 7nfel 2341 . . . 4  |-  F/ x
( F `  z
)  e.  B
9 nfv 1539 . . . 4  |-  F/ z ( F `  x
)  e.  B
10 fveq2 5534 . . . . 5  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
1110eleq1d 2258 . . . 4  |-  ( z  =  x  ->  (
( F `  z
)  e.  B  <->  ( F `  x )  e.  B
) )
122, 3, 8, 9, 11cbvralf 2710 . . 3  |-  ( A. z  e.  A  ( F `  z )  e.  B  <->  A. x  e.  A  ( F `  x )  e.  B )
1312anbi2i 457 . 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 184 1  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    e. wcel 2160   F/_wnfc 2319   A.wral 2468    Fn wfn 5230   -->wf 5231   ` cfv 5235
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243
This theorem is referenced by:  ixpf  6746  cc4f  7298
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