Theorem cmpcov

Description: An open cover of a compact topology has a finite subcover. (Contributed by Jeff Hankins, 29-Jun-2009)

	Ref	Expression
Hypothesis	iscmp.1	|- X = U. J
Assertion	cmpcov	|- ( ( J e. Comp /\ S C_ J /\ X = U. S ) -> E. s e. ( ~P S i^i Fin ) X = U. s )

Proof

Step	Hyp	Ref	Expression
1		iscmp.1	|- X = U. J
2		unieq	|- ( r = S -> U. r = U. S )
3	2	eqeq2d	|- ( r = S -> ( X = U. r <-> X = U. S ) )
4		pweq	|- ( r = S -> ~P r = ~P S )
5	4	ineq1d	|- ( r = S -> ( ~P r i^i Fin ) = ( ~P S i^i Fin ) )
6	5	rexeqdv	|- ( r = S -> ( E. s e. ( ~P r i^i Fin ) X = U. s <-> E. s e. ( ~P S i^i Fin ) X = U. s ) )
7	3 6	imbi12d	|- ( r = S -> ( ( X = U. r -> E. s e. ( ~P r i^i Fin ) X = U. s ) <-> ( X = U. S -> E. s e. ( ~P S i^i Fin ) X = U. s ) ) )
8	1	iscmp	|- ( J e. Comp <-> ( J e. Top /\ A. r e. ~P J ( X = U. r -> E. s e. ( ~P r i^i Fin ) X = U. s ) ) )
9	8	simprbi	|- ( J e. Comp -> A. r e. ~P J ( X = U. r -> E. s e. ( ~P r i^i Fin ) X = U. s ) )
10	9	adantr	|- ( ( J e. Comp /\ S C_ J ) -> A. r e. ~P J ( X = U. r -> E. s e. ( ~P r i^i Fin ) X = U. s ) )
11		ssexg	|- ( ( S C_ J /\ J e. Comp ) -> S e. _V )
12	11	ancoms	|- ( ( J e. Comp /\ S C_ J ) -> S e. _V )
13		simpr	|- ( ( J e. Comp /\ S C_ J ) -> S C_ J )
14	12 13	elpwd	|- ( ( J e. Comp /\ S C_ J ) -> S e. ~P J )
15	7 10 14	rspcdva	|- ( ( J e. Comp /\ S C_ J ) -> ( X = U. S -> E. s e. ( ~P S i^i Fin ) X = U. s ) )
16	15	3impia	|- ( ( J e. Comp /\ S C_ J /\ X = U. S ) -> E. s e. ( ~P S i^i Fin ) X = U. s )