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Theorem wuntpos 10134
 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 10128 . . . . 5 (𝜑 → dom 𝐴𝑈)
41, 3wuncnv 10130 . . . 4 (𝜑dom 𝐴𝑈)
51wun0 10118 . . . . 5 (𝜑 → ∅ ∈ 𝑈)
61, 5wunsn 10116 . . . 4 (𝜑 → {∅} ∈ 𝑈)
71, 4, 6wunun 10110 . . 3 (𝜑 → (dom 𝐴 ∪ {∅}) ∈ 𝑈)
81, 2wunrn 10129 . . 3 (𝜑 → ran 𝐴𝑈)
91, 7, 8wunxp 10124 . 2 (𝜑 → ((dom 𝐴 ∪ {∅}) × ran 𝐴) ∈ 𝑈)
10 tposssxp 7874 . . 3 tpos 𝐴 ⊆ ((dom 𝐴 ∪ {∅}) × ran 𝐴)
1110a1i 11 . 2 (𝜑 → tpos 𝐴 ⊆ ((dom 𝐴 ∪ {∅}) × ran 𝐴))
121, 9, 11wunss 10112 1 (𝜑 → tpos 𝐴𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2114   ∪ cun 3911   ⊆ wss 3913  ∅c0 4269  {csn 4543   × cxp 5529  ◡ccnv 5530  dom cdm 5531  ran crn 5532  tpos ctpos 7869  WUnicwun 10100 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-tpos 7870  df-wun 10102 This theorem is referenced by:  catcoppccl  17347
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