Theorem wunfv 10772

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 10769	. . 3 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈)
4	1, 3	wununi 10746	. 2 ⊢ (𝜑 → ∪ ran 𝐴 ∈ 𝑈)
5		fvssunirn 6939	. . 3 ⊢ (𝐴‘𝐵) ⊆ ∪ ran 𝐴
6	5	a1i 11	. 2 ⊢ (𝜑 → (𝐴‘𝐵) ⊆ ∪ ran 𝐴)
7	1, 4, 6	wunss 10752	1 ⊢ (𝜑 → (𝐴‘𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2108   ⊆ wss 3951  ∪ cuni 4907  ran crn 5686  ‘cfv 6561  WUnicwun 10740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-cnv 5693  df-dm 5695  df-rn 5696  df-iota 6514  df-fv 6569  df-wun 10742

This theorem is referenced by: (None)