Theorem wuntpos

Description: A weak universe is closed under transposition. (Contributed by Mario Carneiro, 12-Jan-2017)

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

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	wundm	$$⊢\phi \to \mathrm{dom}A\in U$$
4	1 3	wuncnv	$$⊢\phi \to {\mathrm{dom}A}^{-1}\in U$$
5	1	wun0	$$⊢\phi \to \varnothing \in U$$
6	1 5	wunsn	$$⊢\phi \to \left\{\varnothing \right\}\in U$$
7	1 4 6	wunun	$$⊢\phi \to {\mathrm{dom}A}^{-1}\cup \left\{\varnothing \right\}\in U$$
8	1 2	wunrn	$$⊢\phi \to \mathrm{ran}A\in U$$
9	1 7 8	wunxp	$$⊢\phi \to \left({\mathrm{dom}A}^{-1}\cup \left\{\varnothing \right\}\right)\times \mathrm{ran}A\in U$$
10		tposssxp	$$⊢tposA\subseteq \left({\mathrm{dom}A}^{-1}\cup \left\{\varnothing \right\}\right)\times \mathrm{ran}A$$
11	10	a1i	$$⊢\phi \to tposA\subseteq \left({\mathrm{dom}A}^{-1}\cup \left\{\varnothing \right\}\right)\times \mathrm{ran}A$$
12	1 9 11	wunss	$$⊢\phi \to tposA\in U$$