Theorem wunf

Description: A weak universe is closed under functions with known domain and codomain. (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 )
	wunf.3	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
Assertion	wunf	⊢ ( 𝜑 → 𝐹 ∈ 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		wunf.3	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
5	1 3 2	wunmap	⊢ ( 𝜑 → ( 𝐵 ↑m 𝐴 ) ∈ 𝑈 )
6	1 5	wunelss	⊢ ( 𝜑 → ( 𝐵 ↑m 𝐴 ) ⊆ 𝑈 )
7	3 2	elmapd	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝐵 ↑m 𝐴 ) ↔ 𝐹 : 𝐴 ⟶ 𝐵 ) )
8	4 7	mpbird	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐵 ↑m 𝐴 ) )
9	6 8	sseldd	⊢ ( 𝜑 → 𝐹 ∈ 𝑈 )