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Theorem dmfex 5628
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 5597 . . 3  |-  ( F : A --> B  ->  dom  F  =  A )
2 dmexg 5132 . . . 4  |-  ( F  e.  C  ->  dom  F  e.  _V )
3 eleq1 2498 . . . 4  |-  ( dom 
F  =  A  -> 
( dom  F  e.  _V 
<->  A  e.  _V )
)
42, 3syl5ib 212 . . 3  |-  ( dom 
F  =  A  -> 
( F  e.  C  ->  A  e.  _V )
)
51, 4syl 16 . 2  |-  ( F : A --> B  -> 
( F  e.  C  ->  A  e.  _V )
)
65impcom 421 1  |-  ( ( F  e.  C  /\  F : A --> B )  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   dom cdm 4880   -->wf 5452
This theorem is referenced by:  wemoiso  6080  fopwdom  7218  fowdom  7541  wdomfil  7944  fin23lem17  8220  fin23lem32  8226  fin23lem39  8232  enfin1ai  8266  fin1a2lem7  8288  kelac1  27140  lindfmm  27276
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-cnv 4888  df-dm 4890  df-rn 4891  df-fn 5459  df-f 5460
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