Theorem wunfv

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

	Ref	Expression
Hypotheses	wun0.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
	wunop.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
Assertion	wunfv	⊢ ( 𝜑 → ( 𝐴 ‘ 𝐵 ) ∈ 𝑈 )

Proof

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