Theorem cvmsrcl

Description: Reverse closure for an even covering. (Contributed by Mario Carneiro, 11-Feb-2015)

	Ref	Expression
Hypothesis	cvmcov.1	$$⊢S=\left(k\in J⟼\left\{s\in \left(𝒫C\setminus \left\{\varnothing \right\}\right)|\left(\bigcup s=F^{-1}\left[k\right]\wedge \forall u\in s\left(\forall v\in \left(s\setminus \left\{u\right\}\right)u\cap v=\varnothing \wedge {F\upharpoonright }_u\in \left(\left(C{\upharpoonright }_{𝑡}u\right)\mathrm{Homeo}\left(J{\upharpoonright }_{𝑡}k\right)\right)\right)\right)\right\}\right)$$
Assertion	cvmsrcl	$$⊢T\in S\left(U\right)\to U\in J$$

Proof

Step	Hyp	Ref	Expression
1		cvmcov.1	$$⊢S=\left(k\in J⟼\left\{s\in \left(𝒫C\setminus \left\{\varnothing \right\}\right)|\left(\bigcup s=F^{-1}\left[k\right]\wedge \forall u\in s\left(\forall v\in \left(s\setminus \left\{u\right\}\right)u\cap v=\varnothing \wedge {F\upharpoonright }_u\in \left(\left(C{\upharpoonright }_{𝑡}u\right)\mathrm{Homeo}\left(J{\upharpoonright }_{𝑡}k\right)\right)\right)\right)\right\}\right)$$
2	1	dmmptss	$$⊢\mathrm{dom}S\subseteq J$$
3		elfvdm	$$⊢T\in S\left(U\right)\to U\in \mathrm{dom}S$$
4	2 3	sseldi	$$⊢T\in S\left(U\right)\to U\in J$$