Theorem wunres 10722

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 5999	. . 3 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ⊆ 𝐴)
5	1, 2, 4	wunss 10703	1 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142   ⊆ wss 3904   ↾ cres 5662  WUnicwun 10691

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-in 3911  df-ss 3921  df-pw 4563  df-uni 4872  df-tr 5218  df-res 5672  df-wun 10693

This theorem is used by:  wunsets  17243