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Theorem wunpm 9869
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 9868 . . 3 (𝜑 → (𝐵 × 𝐴) ∈ 𝑈)
51, 4wunpw 9851 . 2 (𝜑 → 𝒫 (𝐵 × 𝐴) ∈ 𝑈)
6 pmsspw 8162 . . 3 (𝐴pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
76a1i 11 . 2 (𝜑 → (𝐴pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴))
81, 5, 7wunss 9856 1 (𝜑 → (𝐴pm 𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2164  wss 3798  𝒫 cpw 4380   × cxp 5344  (class class class)co 6910  pm cpm 8128  WUnicwun 9844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  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-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-pm 8130  df-wun 9846
This theorem is referenced by:  wunmap  9870  catcfuccl  17118  catcxpccl  17207
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