Theorem wunfv 10692

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 10689	. . 3 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈)
4	1, 3	wununi 10666	. 2 ⊢ (𝜑 → ∪ ran 𝐴 ∈ 𝑈)
5		fvssunirn 6900	. . 3 ⊢ (𝐴‘𝐵) ⊆ ∪ ran 𝐴
6	5	a1i 11	. 2 ⊢ (𝜑 → (𝐴‘𝐵) ⊆ ∪ ran 𝐴)
7	1, 4, 6	wunss 10672	1 ⊢ (𝜑 → (𝐴‘𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2144   ⊆ wss 3906  ∪ cuni 4867  ran crn 5650  ‘cfv 6523  WUnicwun 10660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-cnv 5657  df-dm 5659  df-rn 5660  df-iota 6479  df-fv 6531  df-wun 10662

This theorem is referenced by: (None)