Theorem unidmex 39217

Description: If F is a set, then U. dom F is a set (common case). (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
unidmex.f	|- ( ph -> F e. V )
unidmex.x	|- X = U. dom F

Assertion

Ref	Expression
unidmex	|- ( ph -> X e. _V )

Proof of Theorem unidmex

Step	Hyp	Ref	Expression
1		unidmex.x	. 2 |- X = U. dom F
2		unidmex.f	. . 3 |- ( ph -> F e. V )
3		dmexg 7097	. . 3 |- ( F e. V -> dom F e. _V )
4		uniexg 6955	. . 3 |- ( dom F e. _V -> U. dom F e. _V )
5	2, 3, 4	3syl 18	. 2 |- ( ph -> U. dom F e. _V )
6	1, 5	syl5eqel 2705	1 |- ( ph -> X e. _V )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   U.cuni 4436   dom cdm 5114

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

This theorem is referenced by:  omessle  40712  caragensplit  40714  omeunile  40719  caragenuncl  40727  omeunle  40730  omeiunlempt  40734  carageniuncllem2  40736  caragencmpl  40749