Theorem dmfex 7124

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

Step	Hyp	Ref	Expression
1		fdm 6051	. . 3 |- ( F : A --> B -> dom F = A )
2		dmexg 7097	. . . 4 |- ( F e. C -> dom F e. _V )
3		eleq1 2689	. . . 4 |- ( dom F = A -> ( dom F e. _V <-> A e. _V ) )
4	2, 3	syl5ib 234	. . 3 |- ( dom F = A -> ( F e. C -> A e. _V ) )
5	1, 4	syl 17	. 2 |- ( F : A --> B -> ( F e. C -> A e. _V ) )
6	5	impcom 446	1 |- ( ( F e. C /\ F : A --> B ) -> A e. _V )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   dom cdm 5114   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-fn 5891  df-f 5892

This theorem is referenced by:  wemoiso  7153  fopwdom  8068  fowdom  8476  wdomfil  8884  fin23lem17  9160  fin23lem32  9166  fin23lem39  9172  enfin1ai  9206  fin1a2lem7  9228  lindfmm  20166  kelac1  37633