Theorem wundm

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

	Ref	Expression
Hypotheses	wun0.1	$$⊢\phi \to U\in \mathrm{WUni}$$
	wunop.2	$$⊢\phi \to A\in U$$
Assertion	wundm	$$⊢\phi \to \mathrm{dom}A\in U$$

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	wununi	$$⊢\phi \to \bigcup A\in U$$
4	1 3	wununi	$$⊢\phi \to \bigcup \bigcup A\in U$$
5		ssun1	$$⊢\mathrm{dom}A\subseteq \mathrm{dom}A\cup \mathrm{ran}A$$
6		dmrnssfld	$$⊢\mathrm{dom}A\cup \mathrm{ran}A\subseteq \bigcup \bigcup A$$
7	5 6	sstri	$$⊢\mathrm{dom}A\subseteq \bigcup \bigcup A$$
8	7	a1i	$$⊢\phi \to \mathrm{dom}A\subseteq \bigcup \bigcup A$$
9	1 4 8	wunss	$$⊢\phi \to \mathrm{dom}A\in U$$