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	$$⊢\phi \to U\in \mathrm{WUni}$$
	wunop.2	$$⊢\phi \to A\in U$$
Assertion	wunfv	$$⊢\phi \to A\left(B\right)\in U$$

Proof

Step	Hyp	Ref	Expression
1		wun0.1	$$⊢\phi \to U\in \mathrm{WUni}$$
2		wunop.2	$$⊢\phi \to A\in U$$
3	1 2	wunrn	$$⊢\phi \to \mathrm{ran}A\in U$$
4	1 3	wununi	$$⊢\phi \to \bigcup \mathrm{ran}A\in U$$
5		fvssunirn	$$⊢A\left(B\right)\subseteq \bigcup \mathrm{ran}A$$
6	5	a1i	$$⊢\phi \to A\left(B\right)\subseteq \bigcup \mathrm{ran}A$$
7	1 4 6	wunss	$$⊢\phi \to A\left(B\right)\in U$$