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Theorem unidmex 45055
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 7923 . . 3 (𝐹𝑉 → dom 𝐹 ∈ V)
4 uniexg 7760 . . 3 (dom 𝐹 ∈ V → dom 𝐹 ∈ V)
52, 3, 43syl 18 . 2 (𝜑 dom 𝐹 ∈ V)
61, 5eqeltrid 2845 1 (𝜑𝑋 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480   cuni 4907  dom cdm 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by:  omessle  46513  caragensplit  46515  omeunile  46520  caragenuncl  46528  omeunle  46531  omeiunlempt  46535  carageniuncllem2  46537  caragencmpl  46550
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