Theorem cmpcov 21192

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

Hypothesis

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

Assertion

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

Distinct variable groups:   J, s   S, s

Allowed substitution hint:   X( s)

Proof of Theorem cmpcov

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4444	. . . . 5 |- ( r = S -> U. r = U. S )
2	1	eqeq2d 2632	. . . 4 |- ( r = S -> ( X = U. r <-> X = U. S ) )
3		pweq 4161	. . . . . 6 |- ( r = S -> ~P r = ~P S )
4	3	ineq1d 3813	. . . . 5 |- ( r = S -> ( ~P r i^i Fin ) = ( ~P S i^i Fin ) )
5	4	rexeqdv 3145	. . . 4 |- ( r = S -> ( E. s e. ( ~P r i^i Fin ) X = U. s <-> E. s e. ( ~P S i^i Fin ) X = U. s ) )
6	2, 5	imbi12d 334	. . 3 |- ( 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 ) ) )
7		iscmp.1	. . . . . 6 |- X = U. J
8	7	iscmp 21191	. . . . 5 |- ( 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 480	. . . 4 |- ( 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 481	. . 3 |- ( ( 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		simpr 477	. . . 4 |- ( ( J e. Comp /\ S C_ J ) -> S C_ J )
12		ssexg 4804	. . . . . 6 |- ( ( S C_ J /\ J e. Comp ) -> S e. _V )
13	12	ancoms 469	. . . . 5 |- ( ( J e. Comp /\ S C_ J ) -> S e. _V )
14		elpwg 4166	. . . . 5 |- ( S e. _V -> ( S e. ~P J <-> S C_ J ) )
15	13, 14	syl 17	. . . 4 |- ( ( J e. Comp /\ S C_ J ) -> ( S e. ~P J <-> S C_ J ) )
16	11, 15	mpbird 247	. . 3 |- ( ( J e. Comp /\ S C_ J ) -> S e. ~P J )
17	6, 10, 16	rspcdva 3316	. 2 |- ( ( J e. Comp /\ S C_ J ) -> ( X = U. S -> E. s e. ( ~P S i^i Fin ) X = U. s ) )
18	17	3impia 1261	1 |- ( ( J e. Comp /\ S C_ J /\ X = U. S ) -> E. s e. ( ~P S i^i Fin ) X = U. s )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   Fincfn 7955   Topctop 20698   Compccmp 21189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-cmp 21190

This theorem is referenced by:  cmpcov2  21193  cncmp  21195  discmp  21201  cmpcld  21205  sscmp  21208  comppfsc  21335  alexsubALTlem1  21851  ptcmplem3  21858  lebnum  22763  heibor1  33609