Theorem dmexd 4998

Description: The domain of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
dmexd.1	|- ( ph -> A e. V )

Assertion

Ref	Expression
dmexd	|- ( ph -> dom A e. _V )

Proof of Theorem dmexd

Step	Hyp	Ref	Expression
1		dmexd.1	. 2 |- ( ph -> A e. V )
2		dmexg 4996	. 2 |- ( A e. V -> dom A e. _V )
3	1, 2	syl 14	1 |- ( ph -> dom A e. _V )

Syntax hints:   -> wi 4   e. wcel 2202   _Vcvv 2802   dom cdm 4725

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by:  hashdmprop2dom  11107  usgrsizedgen  16063  wksfval  16172  wlkex  16175