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Theorem wuncnv 10722
Description: A weak universe is closed under the converse operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wun0.1 (𝜑𝑈 ∈ WUni)
wunop.2 (𝜑𝐴𝑈)
Assertion
Ref Expression
wuncnv (𝜑𝐴𝑈)

Proof of Theorem wuncnv
StepHypRef Expression
1 wun0.1 . 2 (𝜑𝑈 ∈ WUni)
2 wunop.2 . . . 4 (𝜑𝐴𝑈)
31, 2wunrn 10721 . . 3 (𝜑 → ran 𝐴𝑈)
41, 2wundm 10720 . . 3 (𝜑 → dom 𝐴𝑈)
51, 3, 4wunxp 10716 . 2 (𝜑 → (ran 𝐴 × dom 𝐴) ∈ 𝑈)
6 cnvssrndm 6261 . . 3 𝐴 ⊆ (ran 𝐴 × dom 𝐴)
76a1i 11 . 2 (𝜑𝐴 ⊆ (ran 𝐴 × dom 𝐴))
81, 5, 7wunss 10704 1 (𝜑𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wss 3941   × cxp 5665  ccnv 5666  dom cdm 5667  ran crn 5668  WUnicwun 10692
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 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-tr 5257  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678  df-wun 10694
This theorem is referenced by:  wuntpos  10726  catcoppccl  18071  catcoppcclOLD  18072
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