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	$$⊢\phi \to U\in \mathrm{WUni}$$
	wunop.2	$$⊢\phi \to A\in U$$
	wunop.3	$$⊢\phi \to B\in U$$
	wunf.3	$$⊢\phi \to F:A⟶B$$
Assertion	wunf	$$⊢\phi \to F\in U$$

Proof

Step	Hyp	Ref	Expression
1		wun0.1	$$⊢\phi \to U\in \mathrm{WUni}$$
2		wunop.2	$$⊢\phi \to A\in U$$
3		wunop.3	$$⊢\phi \to B\in U$$
4		wunf.3	$$⊢\phi \to F:A⟶B$$
5	1 3 2	wunmap	$$⊢\phi \to B^A\in U$$
6	1 5	wunelss	$$⊢\phi \to B^A\subseteq U$$
7	3 2	elmapd	$$⊢\phi \to \left(F\in \left(B^A\right)\leftrightarrow F:A⟶B\right)$$
8	4 7	mpbird	$$⊢\phi \to F\in \left(B^A\right)$$
9	6 8	sseldd	$$⊢\phi \to F\in U$$