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Theorem wundm 9763
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 9741 . . 3 (𝜑 𝐴𝑈)
41, 3wununi 9741 . 2 (𝜑 𝐴𝑈)
5 ssun1 3920 . . . 4 dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6 dmrnssfld 5540 . . . 4 (dom 𝐴 ∪ ran 𝐴) ⊆ 𝐴
75, 6sstri 3754 . . 3 dom 𝐴 𝐴
87a1i 11 . 2 (𝜑 → dom 𝐴 𝐴)
91, 4, 8wunss 9747 1 (𝜑 → dom 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2140  cun 3714  wss 3716   cuni 4589  dom cdm 5267  ran crn 5268  WUnicwun 9735
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 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
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 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-tr 4906  df-cnv 5275  df-dm 5277  df-rn 5278  df-wun 9737
This theorem is referenced by:  wuncnv  9765  wunco  9768  wuntpos  9769  catcoppccl  16980
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