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Theorem unidmex 38702
 Description: If 𝐹 is a set, then ∪ dom 𝐹 is a set (common case). (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 7044 . . 3 (𝐹𝑉 → dom 𝐹 ∈ V)
4 uniexg 6908 . . 3 (dom 𝐹 ∈ V → dom 𝐹 ∈ V)
52, 3, 43syl 18 . 2 (𝜑 dom 𝐹 ∈ V)
61, 5syl5eqel 2702 1 (𝜑𝑋 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987  Vcvv 3186  ∪ cuni 4402  dom cdm 5074 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-cnv 5082  df-dm 5084  df-rn 5085 This theorem is referenced by:  omessle  40019  caragensplit  40021  omeunile  40026  caragenuncl  40034  omeunle  40037  omeiunlempt  40041  carageniuncllem2  40043  caragencmpl  40056
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