Theorem compcov 15429

Description: An open cover of a compact topology has a finite subcover.

Hypothesis

Ref	Expression
compcov.1	|- X = U.J

Assertion

Ref	Expression
compcov	|- ((J e. Comp /\ S C_ J /\ X = U.S) -> E.s e. (~PS i^i Fin)X = U.s)

Distinct variable groups:   J,s   S,s

Proof of Theorem compcov

Step	Hyp	Ref	Expression
1		simpr 350	. . . . 5 |- ((J e. Comp /\ S C_ J) -> S C_ J)
2		ssexg 3457	. . . . . . 7 |- ((S C_ J /\ J e. Comp) -> S e. _V)
3	2	ancoms 484	. . . . . 6 |- ((J e. Comp /\ S C_ J) -> S e. _V)
4		elpwg 3038	. . . . . 6 |- (S e. _V -> (S e. ~PJ <-> S C_ J))
5	3, 4	syl 12	. . . . 5 |- ((J e. Comp /\ S C_ J) -> (S e. ~PJ <-> S C_ J))
6	1, 5	mpbird 213	. . . 4 |- ((J e. Comp /\ S C_ J) -> S e. ~PJ)
7		iscomp 10330	. . . . . 6 |- (J e. Comp <-> (J e. Top /\ A.r e. ~P J(U.J = U.r -> E.s e. (~Pr i^i Fin)U.J = U.s)))
8	7	simprbi 353	. . . . 5 |- (J e. Comp -> A.r e. ~P J(U.J = U.r -> E.s e. (~Pr i^i Fin)U.J = U.s))
9	8	adantr 425	. . . 4 |- ((J e. Comp /\ S C_ J) -> A.r e. ~P J(U.J = U.r -> E.s e. (~Pr i^i Fin)U.J = U.s))
10		unieq 3185	. . . . . . 7 |- (r = S -> U.r = U.S)
11	10	eqeq2d 1895	. . . . . 6 |- (r = S -> (U.J = U.r <-> U.J = U.S))
12		pweq 3036	. . . . . . . 8 |- (r = S -> ~Pr = ~PS)
13	12	ineq1d 2795	. . . . . . 7 |- (r = S -> (~Pr i^i Fin) = (~PS i^i Fin))
14	13	rexeqdv 2270	. . . . . 6 |- (r = S -> (E.s e. (~Pr i^i Fin)U.J = U.s <-> E.s e. (~PS i^i Fin)U.J = U.s))
15	11, 14	imbi12d 688	. . . . 5 |- (r = S -> ((U.J = U.r -> E.s e. (~Pr i^i Fin)U.J = U.s) <-> (U.J = U.S -> E.s e. (~PS i^i Fin)U.J = U.s)))
16	15	rcla4v 2376	. . . 4 |- (S e. ~PJ -> (A.r e. ~P J(U.J = U.r -> E.s e. (~Pr i^i Fin)U.J = U.s) -> (U.J = U.S -> E.s e. (~PS i^i Fin)U.J = U.s)))
17	6, 9, 16	sylc 83	. . 3 |- ((J e. Comp /\ S C_ J) -> (U.J = U.S -> E.s e. (~PS i^i Fin)U.J = U.s))
18		compcov.1	. . . 4 |- X = U.J
19	18	eqeq1i 1891	. . 3 |- (X = U.S <-> U.J = U.S)
20	18	eqeq1i 1891	. . . 4 |- (X = U.s <-> U.J = U.s)
21	20	rexbii 2128	. . 3 |- (E.s e. (~PS i^i Fin)X = U.s <-> E.s e. (~PS i^i Fin)U.J = U.s)
22	17, 19, 21	3imtr4g 612	. 2 |- ((J e. Comp /\ S C_ J) -> (X = U.S -> E.s e. (~PS i^i Fin)X = U.s))
23	22	3impia 1064	1 |- ((J e. Comp /\ S C_ J /\ X = U.S) -> E.s e. (~PS i^i Fin)X = U.s)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292   i^i cin 2592   C_ wss 2593  ~Pcpw 3032  U.cuni 3177  Fincfn 5426  Topctop 8857  Compccomp 10328

This theorem is referenced by:  cptclsscpt 15432  alexsublem1 15437  locfincomp 15514  comppfsc 15517

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-in 2603  df-ss 2605  df-pw 3035  df-uni 3178  df-comp 10329

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