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Theorem dmfex 6648
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 5674 . . 3  |-  ( F : A --> B  ->  dom  F  =  A )
2 dmexg 6622 . . . 4  |-  ( F  e.  C  ->  dom  F  e.  _V )
3 eleq1 2526 . . . 4  |-  ( dom 
F  =  A  -> 
( dom  F  e.  _V 
<->  A  e.  _V )
)
42, 3syl5ib 219 . . 3  |-  ( dom 
F  =  A  -> 
( F  e.  C  ->  A  e.  _V )
)
51, 4syl 16 . 2  |-  ( F : A --> B  -> 
( F  e.  C  ->  A  e.  _V )
)
65impcom 430 1  |-  ( ( F  e.  C  /\  F : A --> B )  ->  A  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   dom cdm 4951   -->wf 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-cnv 4959  df-dm 4961  df-rn 4962  df-fn 5532  df-f 5533
This theorem is referenced by:  wemoiso  6675  fopwdom  7532  fowdom  7900  wdomfil  8345  fin23lem17  8621  fin23lem32  8627  fin23lem39  8633  enfin1ai  8667  fin1a2lem7  8689  lindfmm  18384  kelac1  29584
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