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Theorem unidmex 41608
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 7608 . . 3 (𝐹𝑉 → dom 𝐹 ∈ V)
4 uniexg 7460 . . 3 (dom 𝐹 ∈ V → dom 𝐹 ∈ V)
52, 3, 43syl 18 . 2 (𝜑 dom 𝐹 ∈ V)
61, 5eqeltrid 2920 1 (𝜑𝑋 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3480   cuni 4824  dom cdm 5542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-cnv 5550  df-dm 5552  df-rn 5553
This theorem is referenced by:  omessle  43067  caragensplit  43069  omeunile  43074  caragenuncl  43082  omeunle  43085  omeiunlempt  43089  carageniuncllem2  43091  caragencmpl  43104
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