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Theorem wunres 10732
Description: A weak universe is closed under restrictions. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
wun0.1 (𝜑𝑈 ∈ WUni)
wunop.2 (𝜑𝐴𝑈)
Assertion
Ref Expression
wunres (𝜑 → (𝐴𝐵) ∈ 𝑈)

Proof of Theorem wunres
StepHypRef Expression
1 wun0.1 . 2 (𝜑𝑈 ∈ WUni)
2 wunop.2 . 2 (𝜑𝐴𝑈)
3 resss 6006 . . 3 (𝐴𝐵) ⊆ 𝐴
43a1i 11 . 2 (𝜑 → (𝐴𝐵) ⊆ 𝐴)
51, 2, 4wunss 10713 1 (𝜑 → (𝐴𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wss 3948  cres 5678  WUnicwun 10701
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1088  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-in 3955  df-ss 3965  df-pw 4604  df-uni 4909  df-tr 5266  df-res 5688  df-wun 10703
This theorem is referenced by:  wunsets  17117
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