Theorem dmfex 5079

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 5050	. . 3 |- ( F : A --> B -> dom F = A )
2		dmexg 4596	. . . 4 |- ( F e. C -> dom F e. _V )
3		eleq1 2100	. . . 4 |- ( dom F = A -> ( dom F e. _V <-> A e. _V ) )
4	2, 3	syl5ib 143	. . 3 |- ( dom F = A -> ( F e. C -> A e. _V ) )
5	1, 4	syl 14	. 2 |- ( F : A --> B -> ( F e. C -> A e. _V ) )
6	5	impcom 116	1 |- ( ( F e. C /\ F : A --> B ) -> A e. _V )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   _Vcvv 2557   dom cdm 4345   -->wf 4898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356  df-fn 4905  df-f 4906

This theorem is referenced by:  fopwdom  6310