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Theorem wunpm 10136
Description: A weak universe is closed under partial mappings. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wun0.1 (𝜑𝑈 ∈ WUni)
wunop.2 (𝜑𝐴𝑈)
wunop.3 (𝜑𝐵𝑈)
Assertion
Ref Expression
wunpm (𝜑 → (𝐴pm 𝐵) ∈ 𝑈)

Proof of Theorem wunpm
StepHypRef Expression
1 wun0.1 . 2 (𝜑𝑈 ∈ WUni)
2 wunop.3 . . . 4 (𝜑𝐵𝑈)
3 wunop.2 . . . 4 (𝜑𝐴𝑈)
41, 2, 3wunxp 10135 . . 3 (𝜑 → (𝐵 × 𝐴) ∈ 𝑈)
51, 4wunpw 10118 . 2 (𝜑 → 𝒫 (𝐵 × 𝐴) ∈ 𝑈)
6 pmsspw 8431 . . 3 (𝐴pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
76a1i 11 . 2 (𝜑 → (𝐴pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴))
81, 5, 7wunss 10123 1 (𝜑 → (𝐴pm 𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wss 3935  𝒫 cpw 4537   × cxp 5547  (class class class)co 7145  pm cpm 8397  WUnicwun 10111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7680  df-2nd 7681  df-pm 8399  df-wun 10113
This theorem is referenced by:  wunmap  10137  catcfuccl  17359  catcxpccl  17447
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