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Theorem unidmex 39736
Description: If 𝐹 is a set, then dom 𝐹 is a set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
unidmex.f (𝜑𝐹𝑉)
unidmex.x 𝑋 = dom 𝐹
Assertion
Ref Expression
unidmex (𝜑𝑋 ∈ V)

Proof of Theorem unidmex
StepHypRef Expression
1 unidmex.x . 2 𝑋 = dom 𝐹
2 unidmex.f . . 3 (𝜑𝐹𝑉)
3 dmexg 7248 . . 3 (𝐹𝑉 → dom 𝐹 ∈ V)
4 uniexg 7106 . . 3 (dom 𝐹 ∈ V → dom 𝐹 ∈ V)
52, 3, 43syl 18 . 2 (𝜑 dom 𝐹 ∈ V)
61, 5syl5eqel 2854 1 (𝜑𝑋 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  Vcvv 3351   cuni 4575  dom cdm 5250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-cnv 5258  df-dm 5260  df-rn 5261
This theorem is referenced by:  omessle  41227  caragensplit  41229  omeunile  41234  caragenuncl  41242  omeunle  41245  omeiunlempt  41249  carageniuncllem2  41251  caragencmpl  41264
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