Theorem wunmap

Description: A weak universe is closed under mappings. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypotheses	wun0.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
	wunop.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
	wunop.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑈 )
Assertion	wunmap	⊢ ( 𝜑 → ( 𝐴 ↑m 𝐵 ) ∈ 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		wun0.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
2		wunop.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
3		wunop.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑈 )
4	1 2 3	wunpm	⊢ ( 𝜑 → ( 𝐴 ↑pm 𝐵 ) ∈ 𝑈 )
5		mapsspm	⊢ ( 𝐴 ↑m 𝐵 ) ⊆ ( 𝐴 ↑pm 𝐵 )
6	5	a1i	⊢ ( 𝜑 → ( 𝐴 ↑m 𝐵 ) ⊆ ( 𝐴 ↑pm 𝐵 ) )
7	1 4 6	wunss	⊢ ( 𝜑 → ( 𝐴 ↑m 𝐵 ) ∈ 𝑈 )