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Theorem ffnfvf 6047
Description: A function maps to a class to which all values belong. This version of ffnfv 6046 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 6046 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. z  e.  A  ( F `  z )  e.  B
) )
2 nfcv 2629 . . . 4  |-  F/_ z A
3 ffnfvf.1 . . . 4  |-  F/_ x A
4 ffnfvf.3 . . . . . 6  |-  F/_ x F
5 nfcv 2629 . . . . . 6  |-  F/_ x
z
64, 5nffv 5872 . . . . 5  |-  F/_ x
( F `  z
)
7 ffnfvf.2 . . . . 5  |-  F/_ x B
86, 7nfel 2642 . . . 4  |-  F/ x
( F `  z
)  e.  B
9 nfv 1683 . . . 4  |-  F/ z ( F `  x
)  e.  B
10 fveq2 5865 . . . . 5  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
1110eleq1d 2536 . . . 4  |-  ( z  =  x  ->  (
( F `  z
)  e.  B  <->  ( F `  x )  e.  B
) )
122, 3, 8, 9, 11cbvralf 3082 . . 3  |-  ( A. z  e.  A  ( F `  z )  e.  B  <->  A. x  e.  A  ( F `  x )  e.  B )
1312anbi2i 694 . 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 249 1  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    e. wcel 1767   F/_wnfc 2615   A.wral 2814    Fn wfn 5582   -->wf 5583   ` cfv 5587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-fv 5595
This theorem is referenced by:  ixpf  7491
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