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Theorem wundm 10672
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 10650 . . 3 (𝜑 𝐴𝑈)
41, 3wununi 10650 . 2 (𝜑 𝐴𝑈)
5 ssun1 4136 . . . 4 dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6 dmrnssfld 5929 . . . 4 (dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴
75, 6sstri 3957 . . 3 dom 𝐴 𝐴
87a1i 11 . 2 (𝜑 → dom 𝐴 𝐴)
91, 4, 8wunss 10656 1 (𝜑 → dom 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  cun 3912  wss 3914   cuni 4869  dom cdm 5637  ran crn 5638  WUnicwun 10644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-tr 5227  df-cnv 5645  df-dm 5647  df-rn 5648  df-wun 10646
This theorem is referenced by:  wuncnv  10674  wunco  10677  wuntpos  10678  catcoppccl  18011  catcoppcclOLD  18012
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