Theorem wunfv 10647

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 10644	. . 3 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈)
4	1, 3	wununi 10621	. 2 ⊢ (𝜑 → ∪ ran 𝐴 ∈ 𝑈)
5		fvssunirn 6866	. . 3 ⊢ (𝐴‘𝐵) ⊆ ∪ ran 𝐴
6	5	a1i 11	. 2 ⊢ (𝜑 → (𝐴‘𝐵) ⊆ ∪ ran 𝐴)
7	1, 4, 6	wunss 10627	1 ⊢ (𝜑 → (𝐴‘𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2114   ⊆ wss 3902  ∪ cuni 4864  ran crn 5626  ‘cfv 6493  WUnicwun 10615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-tr 5207  df-cnv 5633  df-dm 5635  df-rn 5636  df-iota 6449  df-fv 6501  df-wun 10617

This theorem is referenced by: (None)