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Theorem wundm 10753
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 10731 . . 3 (𝜑 𝐴𝑈)
41, 3wununi 10731 . 2 (𝜑 𝐴𝑈)
5 ssun1 4170 . . . 4 dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6 dmrnssfld 5973 . . . 4 (dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴
75, 6sstri 3986 . . 3 dom 𝐴 𝐴
87a1i 11 . 2 (𝜑 → dom 𝐴 𝐴)
91, 4, 8wunss 10737 1 (𝜑 → dom 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  cun 3942  wss 3944   cuni 4909  dom cdm 5678  ran crn 5679  WUnicwun 10725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-cnv 5686  df-dm 5688  df-rn 5689  df-wun 10727
This theorem is referenced by:  wuncnv  10755  wunco  10758  wuntpos  10759  catcoppccl  18109  catcoppcclOLD  18110
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