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Theorem dmfex 5462
Description: If a mapping is a set, its domain is a set. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
dmfex  |-  ( ( F  e.  C  /\  F : A --> B )  ->  A  e.  _V )

Proof of Theorem dmfex
StepHypRef Expression
1 fdm 5431 . . 3  |-  ( F : A --> B  ->  dom  F  =  A )
2 dmexg 4976 . . . 4  |-  ( F  e.  C  ->  dom  F  e.  _V )
3 eleq1 2376 . . . 4  |-  ( dom 
F  =  A  -> 
( dom  F  e.  _V 
<->  A  e.  _V )
)
42, 3syl5ib 210 . . 3  |-  ( dom 
F  =  A  -> 
( F  e.  C  ->  A  e.  _V )
)
51, 4syl 15 . 2  |-  ( F : A --> B  -> 
( F  e.  C  ->  A  e.  _V )
)
65impcom 419 1  |-  ( ( F  e.  C  /\  F : A --> B )  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   _Vcvv 2822   dom cdm 4726   -->wf 5288
This theorem is referenced by:  wemoiso  5897  fopwdom  7013  fowdom  7330  wdomfil  7733  fin23lem17  8009  fin23lem32  8015  fin23lem39  8021  enfin1ai  8055  fin1a2lem7  8077  kelac1  26309  lindfmm  26445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-cnv 4734  df-dm 4736  df-rn 4737  df-fn 5295  df-f 5296
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