Theorem cmpfiiin 37260

Description: In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Hypotheses

Ref	Expression
cmpfiiin.x	|- X = U. J
cmpfiiin.j	|- ( ph -> J e. Comp )
cmpfiiin.s	|- ( ( ph /\ k e. I ) -> S e. ( Clsd ` J ) )
cmpfiiin.z	|- ( ( ph /\ ( l C_ I /\ l e. Fin ) ) -> ( X i^i |^|_ k e. l S ) =/= (/) )

Assertion

Ref	Expression
cmpfiiin	|- ( ph -> ( X i^i |^|_ k e. I S ) =/= (/) )

Distinct variable groups:   ph, k, l   k, I, l   k, J, l   S, l   k, X, l

Allowed substitution hint:   S( k)

Proof of Theorem cmpfiiin

Step	Hyp	Ref	Expression
1		cmpfiiin.j	. . . . 5 |- ( ph -> J e. Comp )
2		cmptop 21198	. . . . 5 |- ( J e. Comp -> J e. Top )
3	1, 2	syl 17	. . . 4 |- ( ph -> J e. Top )
4		cmpfiiin.x	. . . . 5 |- X = U. J
5	4	topcld 20839	. . . 4 |- ( J e. Top -> X e. ( Clsd ` J ) )
6	3, 5	syl 17	. . 3 |- ( ph -> X e. ( Clsd ` J ) )
7		cmpfiiin.s	. . . . 5 |- ( ( ph /\ k e. I ) -> S e. ( Clsd ` J ) )
8	4	cldss 20833	. . . . 5 |- ( S e. ( Clsd ` J ) -> S C_ X )
9	7, 8	syl 17	. . . 4 |- ( ( ph /\ k e. I ) -> S C_ X )
10	9	ralrimiva 2966	. . 3 |- ( ph -> A. k e. I S C_ X )
11		riinint 5382	. . 3 |- ( ( X e. ( Clsd ` J ) /\ A. k e. I S C_ X ) -> ( X i^i |^|_ k e. I S ) = |^| ( { X } u. ran ( k e. I |-> S ) ) )
12	6, 10, 11	syl2anc 693	. 2 |- ( ph -> ( X i^i |^|_ k e. I S ) = |^| ( { X } u. ran ( k e. I |-> S ) ) )
13	6	snssd 4340	. . . 4 |- ( ph -> { X } C_ ( Clsd ` J ) )
14		eqid 2622	. . . . . 6 |- ( k e. I |-> S ) = ( k e. I |-> S )
15	7, 14	fmptd 6385	. . . . 5 |- ( ph -> ( k e. I |-> S ) : I --> ( Clsd ` J ) )
16		frn 6053	. . . . 5 |- ( ( k e. I |-> S ) : I --> ( Clsd ` J ) -> ran ( k e. I |-> S ) C_ ( Clsd ` J ) )
17	15, 16	syl 17	. . . 4 |- ( ph -> ran ( k e. I |-> S ) C_ ( Clsd ` J ) )
18	13, 17	unssd 3789	. . 3 |- ( ph -> ( { X } u. ran ( k e. I |-> S ) ) C_ ( Clsd ` J ) )
19		elin 3796	. . . . . . 7 |- ( l e. ( ~P I i^i Fin ) <-> ( l e. ~P I /\ l e. Fin ) )
20		elpwi 4168	. . . . . . . 8 |- ( l e. ~P I -> l C_ I )
21	20	anim1i 592	. . . . . . 7 |- ( ( l e. ~P I /\ l e. Fin ) -> ( l C_ I /\ l e. Fin ) )
22	19, 21	sylbi 207	. . . . . 6 |- ( l e. ( ~P I i^i Fin ) -> ( l C_ I /\ l e. Fin ) )
23		cmpfiiin.z	. . . . . . 7 |- ( ( ph /\ ( l C_ I /\ l e. Fin ) ) -> ( X i^i |^|_ k e. l S ) =/= (/) )
24		nesym 2850	. . . . . . 7 |- ( ( X i^i |^|_ k e. l S ) =/= (/) <-> -. (/) = ( X i^i |^|_ k e. l S ) )
25	23, 24	sylib 208	. . . . . 6 |- ( ( ph /\ ( l C_ I /\ l e. Fin ) ) -> -. (/) = ( X i^i |^|_ k e. l S ) )
26	22, 25	sylan2 491	. . . . 5 |- ( ( ph /\ l e. ( ~P I i^i Fin ) ) -> -. (/) = ( X i^i |^|_ k e. l S ) )
27	26	nrexdv 3001	. . . 4 |- ( ph -> -. E. l e. ( ~P I i^i Fin ) (/) = ( X i^i |^|_ k e. l S ) )
28		elrfirn2 37259	. . . . 5 |- ( ( X e. ( Clsd ` J ) /\ A. k e. I S C_ X ) -> ( (/) e. ( fi ` ( { X } u. ran ( k e. I |-> S ) ) ) <-> E. l e. ( ~P I i^i Fin ) (/) = ( X i^i |^|_ k e. l S ) ) )
29	6, 10, 28	syl2anc 693	. . . 4 |- ( ph -> ( (/) e. ( fi ` ( { X } u. ran ( k e. I |-> S ) ) ) <-> E. l e. ( ~P I i^i Fin ) (/) = ( X i^i |^|_ k e. l S ) ) )
30	27, 29	mtbird 315	. . 3 |- ( ph -> -. (/) e. ( fi ` ( { X } u. ran ( k e. I |-> S ) ) ) )
31		cmpfii 21212	. . 3 |- ( ( J e. Comp /\ ( { X } u. ran ( k e. I |-> S ) ) C_ ( Clsd ` J ) /\ -. (/) e. ( fi ` ( { X } u. ran ( k e. I |-> S ) ) ) ) -> |^| ( { X } u. ran ( k e. I |-> S ) ) =/= (/) )
32	1, 18, 30, 31	syl3anc 1326	. 2 |- ( ph -> |^| ( { X } u. ran ( k e. I |-> S ) ) =/= (/) )
33	12, 32	eqnetrd 2861	1 |- ( ph -> ( X i^i |^|_ k e. I S ) =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   |^|cint 4475   |^|_ciin 4521   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888   Fincfn 7955   ficfi 8316   Topctop 20698   Clsdccld 20820   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-top 20699  df-cld 20823  df-cmp 21190

This theorem is referenced by:  kelac1  37633