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Theorem wuncnv 10141
 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 10140 . . 3 (𝜑 → ran 𝐴𝑈)
41, 2wundm 10139 . . 3 (𝜑 → dom 𝐴𝑈)
51, 3, 4wunxp 10135 . 2 (𝜑 → (ran 𝐴 × dom 𝐴) ∈ 𝑈)
6 cnvssrndm 6100 . . 3 𝐴 ⊆ (ran 𝐴 × dom 𝐴)
76a1i 11 . 2 (𝜑𝐴 ⊆ (ran 𝐴 × dom 𝐴))
81, 5, 7wunss 10123 1 (𝜑𝐴𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2114   ⊆ wss 3908   × cxp 5530  ◡ccnv 5531  dom cdm 5532  ran crn 5533  WUnicwun 10111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-tr 5149  df-xp 5538  df-rel 5539  df-cnv 5540  df-dm 5542  df-rn 5543  df-wun 10113 This theorem is referenced by:  wuntpos  10145  catcoppccl  17359
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