Theorem dmfex 5535

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 5495	. . 3 |- ( F : A --> B -> dom F = A )
2		dmexg 5002	. . . 4 |- ( F e. C -> dom F e. _V )
3		eleq1 2294	. . . 4 |- ( dom F = A -> ( dom F e. _V <-> A e. _V ) )
4	2, 3	imbitrid 154	. . 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 125	1 |- ( ( F e. C /\ F : A --> B ) -> A e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   dom cdm 4731   -->wf 5329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-cnv 4739  df-dm 4741  df-rn 4742  df-fn 5336  df-f 5337

This theorem is referenced by:  fopwdom  7065