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Theorem wuntpos 9871
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 9865 . . . . 5 (𝜑 → dom 𝐴𝑈)
41, 3wuncnv 9867 . . . 4 (𝜑dom 𝐴𝑈)
51wun0 9855 . . . . 5 (𝜑 → ∅ ∈ 𝑈)
61, 5wunsn 9853 . . . 4 (𝜑 → {∅} ∈ 𝑈)
71, 4, 6wunun 9847 . . 3 (𝜑 → (dom 𝐴 ∪ {∅}) ∈ 𝑈)
81, 2wunrn 9866 . . 3 (𝜑 → ran 𝐴𝑈)
91, 7, 8wunxp 9861 . 2 (𝜑 → ((dom 𝐴 ∪ {∅}) × ran 𝐴) ∈ 𝑈)
10 tposssxp 7621 . . 3 tpos 𝐴 ⊆ ((dom 𝐴 ∪ {∅}) × ran 𝐴)
1110a1i 11 . 2 (𝜑 → tpos 𝐴 ⊆ ((dom 𝐴 ∪ {∅}) × ran 𝐴))
121, 9, 11wunss 9849 1 (𝜑 → tpos 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2166  cun 3796  wss 3798  c0 4144  {csn 4397   × cxp 5340  ccnv 5341  dom cdm 5342  ran crn 5343  tpos ctpos 7616  WUnicwun 9837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-tpos 7617  df-wun 9839
This theorem is referenced by:  catcoppccl  17110
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