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Theorem unidmex 39716
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 7262 . . 3 (𝐹𝑉 → dom 𝐹 ∈ V)
4 uniexg 7120 . . 3 (dom 𝐹 ∈ V → dom 𝐹 ∈ V)
52, 3, 43syl 18 . 2 (𝜑 dom 𝐹 ∈ V)
61, 5syl5eqel 2843 1 (𝜑𝑋 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340   cuni 4588  dom cdm 5266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-cnv 5274  df-dm 5276  df-rn 5277
This theorem is referenced by:  omessle  41218  caragensplit  41220  omeunile  41225  caragenuncl  41233  omeunle  41236  omeiunlempt  41240  carageniuncllem2  41242  caragencmpl  41255
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