Theorem cvmsuni

Description: An even covering of U has union equal to the preimage of U by F . (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	cvmsuni	$$⊢T\in S\left(U\right)\to \bigcup T=F^{-1}\left[U\right]$$

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	cvmsi	$$⊢T\in S\left(U\right)\to \left(U\in J\wedge \left(T\subseteq C\wedge T\ne \varnothing \right)\wedge \left(\bigcup T=F^{-1}\left[U\right]\wedge \forall u\in T\left(\forall v\in \left(T\setminus \left\{u\right\}\right)u\cap v=\varnothing \wedge {F\upharpoonright }_u\in \left(\left(C{\upharpoonright }_{𝑡}u\right)\mathrm{Homeo}\left(J{\upharpoonright }_{𝑡}U\right)\right)\right)\right)\right)$$
3	2	simp3d	$$⊢T\in S\left(U\right)\to \left(\bigcup T=F^{-1}\left[U\right]\wedge \forall u\in T\left(\forall v\in \left(T\setminus \left\{u\right\}\right)u\cap v=\varnothing \wedge {F\upharpoonright }_u\in \left(\left(C{\upharpoonright }_{𝑡}u\right)\mathrm{Homeo}\left(J{\upharpoonright }_{𝑡}U\right)\right)\right)\right)$$
4	3	simpld	$$⊢T\in S\left(U\right)\to \bigcup T=F^{-1}\left[U\right]$$