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Theorem wundm 10768
Description: A weak universe is closed under the domain operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wun0.1 (𝜑𝑈 ∈ WUni)
wunop.2 (𝜑𝐴𝑈)
Assertion
Ref Expression
wundm (𝜑 → dom 𝐴𝑈)

Proof of Theorem wundm
StepHypRef Expression
1 wun0.1 . 2 (𝜑𝑈 ∈ WUni)
2 wunop.2 . . . 4 (𝜑𝐴𝑈)
31, 2wununi 10746 . . 3 (𝜑 𝐴𝑈)
41, 3wununi 10746 . 2 (𝜑 𝐴𝑈)
5 ssun1 4178 . . . 4 dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6 dmrnssfld 5984 . . . 4 (dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴
75, 6sstri 3993 . . 3 dom 𝐴 𝐴
87a1i 11 . 2 (𝜑 → dom 𝐴 𝐴)
91, 4, 8wunss 10752 1 (𝜑 → dom 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  cun 3949  wss 3951   cuni 4907  dom cdm 5685  ran crn 5686  WUnicwun 10740
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
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-ne 2941  df-ral 3062  df-rex 3071  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-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-cnv 5693  df-dm 5695  df-rn 5696  df-wun 10742
This theorem is referenced by:  wuncnv  10770  wunco  10773  wuntpos  10774  catcoppccl  18162
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