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Theorem wuntpos 10764
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 10758 . . . . 5 (𝜑 → dom 𝐴𝑈)
41, 3wuncnv 10760 . . . 4 (𝜑dom 𝐴𝑈)
51wun0 10748 . . . . 5 (𝜑 → ∅ ∈ 𝑈)
61, 5wunsn 10746 . . . 4 (𝜑 → {∅} ∈ 𝑈)
71, 4, 6wunun 10740 . . 3 (𝜑 → (dom 𝐴 ∪ {∅}) ∈ 𝑈)
81, 2wunrn 10759 . . 3 (𝜑 → ran 𝐴𝑈)
91, 7, 8wunxp 10754 . 2 (𝜑 → ((dom 𝐴 ∪ {∅}) × ran 𝐴) ∈ 𝑈)
10 tposssxp 8236 . . 3 tpos 𝐴 ⊆ ((dom 𝐴 ∪ {∅}) × ran 𝐴)
1110a1i 11 . 2 (𝜑 → tpos 𝐴 ⊆ ((dom 𝐴 ∪ {∅}) × ran 𝐴))
121, 9, 11wunss 10742 1 (𝜑 → tpos 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  cun 3942  wss 3944  c0 4322  {csn 4630   × cxp 5676  ccnv 5677  dom cdm 5678  ran crn 5679  tpos ctpos 8231  WUnicwun 10730
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-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-tpos 8232  df-wun 10732
This theorem is referenced by:  catcoppccl  18125  catcoppcclOLD  18126
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