Theorem wunfv 10724

Description: A weak universe is closed under the function value operator. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
wunfv	⊢ (𝜑 → (𝐴‘𝐵) ∈ 𝑈)

Proof of Theorem wunfv

Step	Hyp	Ref	Expression
1		wun0.1	. 2 ⊢ (𝜑 → 𝑈 ∈ WUni)
2		wunop.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
3	1, 2	wunrn 10721	. . 3 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈)
4	1, 3	wununi 10698	. 2 ⊢ (𝜑 → ∪ ran 𝐴 ∈ 𝑈)
5		fvssunirn 6915	. . 3 ⊢ (𝐴‘𝐵) ⊆ ∪ ran 𝐴
6	5	a1i 11	. 2 ⊢ (𝜑 → (𝐴‘𝐵) ⊆ ∪ ran 𝐴)
7	1, 4, 6	wunss 10704	1 ⊢ (𝜑 → (𝐴‘𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2098   ⊆ wss 3941  ∪ cuni 4900  ran crn 5668  ‘cfv 6534  WUnicwun 10692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-tr 5257  df-cnv 5675  df-dm 5677  df-rn 5678  df-iota 6486  df-fv 6542  df-wun 10694

This theorem is referenced by: (None)