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Theorem wuncnv 10748
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 10747 . . 3 (𝜑 → ran 𝐴𝑈)
41, 2wundm 10746 . . 3 (𝜑 → dom 𝐴𝑈)
51, 3, 4wunxp 10742 . 2 (𝜑 → (ran 𝐴 × dom 𝐴) ∈ 𝑈)
6 cnvssrndm 6270 . . 3 𝐴 ⊆ (ran 𝐴 × dom 𝐴)
76a1i 11 . 2 (𝜑𝐴 ⊆ (ran 𝐴 × dom 𝐴))
81, 5, 7wunss 10730 1 (𝜑𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  wss 3945   × cxp 5671  ccnv 5672  dom cdm 5673  ran crn 5674  WUnicwun 10718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-tr 5261  df-xp 5679  df-rel 5680  df-cnv 5681  df-dm 5683  df-rn 5684  df-wun 10720
This theorem is referenced by:  wuntpos  10752  catcoppccl  18100  catcoppcclOLD  18101
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