ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmfex Unicode version

Theorem dmfex 5235
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 5201 . . 3  |-  ( F : A --> B  ->  dom  F  =  A )
2 dmexg 4729 . . . 4  |-  ( F  e.  C  ->  dom  F  e.  _V )
3 eleq1 2157 . . . 4  |-  ( dom 
F  =  A  -> 
( dom  F  e.  _V 
<->  A  e.  _V )
)
42, 3syl5ib 153 . . 3  |-  ( dom 
F  =  A  -> 
( F  e.  C  ->  A  e.  _V )
)
51, 4syl 14 . 2  |-  ( F : A --> B  -> 
( F  e.  C  ->  A  e.  _V )
)
65impcom 124 1  |-  ( ( F  e.  C  /\  F : A --> B )  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1296    e. wcel 1445   _Vcvv 2633   dom cdm 4467   -->wf 5045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-cnv 4475  df-dm 4477  df-rn 4478  df-fn 5052  df-f 5053
This theorem is referenced by:  fopwdom  6632
  Copyright terms: Public domain W3C validator