Theorem wunres 10487

Description: A weak universe is closed under restrictions. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
wun0.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wunop.2	⊢ (𝜑 → 𝐴 ∈ 𝑈)

Assertion

Ref	Expression
wunres	⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ 𝑈)

Proof of Theorem wunres

Step	Hyp	Ref	Expression
1		wun0.1	. 2 ⊢ (𝜑 → 𝑈 ∈ WUni)
2		wunop.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
3		resss 5916	. . 3 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ⊆ 𝐴)
5	1, 2, 4	wunss 10468	1 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2106   ⊆ wss 3887   ↾ cres 5591  WUnicwun 10456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-pw 4535  df-uni 4840  df-tr 5192  df-res 5601  df-wun 10458

This theorem is referenced by:  wunsets  16878