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Theorem wuntpos 10772
Description: A weak universe is closed under transposition. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
wun0.1 (𝜑𝑈 ∈ WUni)
wunop.2 (𝜑𝐴𝑈)
Assertion
Ref Expression
wuntpos (𝜑 → tpos 𝐴𝑈)

Proof of Theorem wuntpos
StepHypRef Expression
1 wun0.1 . 2 (𝜑𝑈 ∈ WUni)
2 wunop.2 . . . . . 6 (𝜑𝐴𝑈)
31, 2wundm 10766 . . . . 5 (𝜑 → dom 𝐴𝑈)
41, 3wuncnv 10768 . . . 4 (𝜑dom 𝐴𝑈)
51wun0 10756 . . . . 5 (𝜑 → ∅ ∈ 𝑈)
61, 5wunsn 10754 . . . 4 (𝜑 → {∅} ∈ 𝑈)
71, 4, 6wunun 10748 . . 3 (𝜑 → (dom 𝐴 ∪ {∅}) ∈ 𝑈)
81, 2wunrn 10767 . . 3 (𝜑 → ran 𝐴𝑈)
91, 7, 8wunxp 10762 . 2 (𝜑 → ((dom 𝐴 ∪ {∅}) × ran 𝐴) ∈ 𝑈)
10 tposssxp 8254 . . 3 tpos 𝐴 ⊆ ((dom 𝐴 ∪ {∅}) × ran 𝐴)
1110a1i 11 . 2 (𝜑 → tpos 𝐴 ⊆ ((dom 𝐴 ∪ {∅}) × ran 𝐴))
121, 9, 11wunss 10750 1 (𝜑 → tpos 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cun 3961  wss 3963  c0 4339  {csn 4631   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  tpos ctpos 8249  WUnicwun 10738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-tpos 8250  df-wun 10740
This theorem is referenced by:  catcoppccl  18171  catcoppcclOLD  18172
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