Theorem wunres 9432

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 5342	. . 3 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ⊆ 𝐴)
5	1, 2, 4	wunss 9413	1 ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 1977   ⊆ wss 3540   ↾ cres 5040  WUnicwun 9401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373  df-tr 4681  df-res 5050  df-wun 9403

This theorem is referenced by:  wunsets  15728