Theorem wunres 10787

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 5988	. . 3 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ⊆ 𝐴)
5	1, 2, 4	wunss 10768	1 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ⊆ wss 3898   ↾ cres 5649  WUnicwun 10756

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-in 3905  df-ss 3915  df-pw 4558  df-uni 4867  df-tr 5212  df-res 5659  df-wun 10758

This theorem is used by:  wunsets  17316