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Theorem dmfex 5424
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 5393 . . 3  |-  ( F : A --> B  ->  dom  F  =  A )
2 dmexg 4939 . . . 4  |-  ( F  e.  C  ->  dom  F  e.  _V )
3 eleq1 2343 . . . 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 1623    e. wcel 1684   _Vcvv 2788   dom cdm 4689   -->wf 5251
This theorem is referenced by:  wemoiso  5855  fopwdom  6970  fowdom  7285  wdomfil  7688  fin23lem17  7964  fin23lem32  7970  fin23lem39  7976  enfin1ai  8010  fin1a2lem7  8032  kelac1  27161  lindfmm  27297
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-13 1686  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  ax-un 4512
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-rex 2549  df-rab 2552  df-v 2790  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-cnv 4697  df-dm 4699  df-rn 4700  df-fn 5258  df-f 5259
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