Theorem fclscmpi 21833

Description: Forward direction of fclscmp 21834. Every filter clusters in a compact space. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Hypothesis

Ref	Expression
flimfnfcls.x	|- X = U. J

Assertion

Ref	Expression
fclscmpi	|- ( ( J e. Comp /\ F e. ( Fil ` X ) ) -> ( J fClus F ) =/= (/) )

Proof of Theorem fclscmpi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cmptop 21198	. . . 4 |- ( J e. Comp -> J e. Top )
2		flimfnfcls.x	. . . . . 6 |- X = U. J
3	2	fclsval 21812	. . . . 5 |- ( ( J e. Top /\ F e. ( Fil ` X ) ) -> ( J fClus F ) = if ( X = X , |^|_ x e. F ( ( cls ` J ) ` x ) , (/) ) )
4		eqid 2622	. . . . . 6 |- X = X
5	4	iftruei 4093	. . . . 5 |- if ( X = X , |^|_ x e. F ( ( cls ` J ) ` x ) , (/) ) = |^|_ x e. F ( ( cls ` J ) ` x )
6	3, 5	syl6eq 2672	. . . 4 |- ( ( J e. Top /\ F e. ( Fil ` X ) ) -> ( J fClus F ) = |^|_ x e. F ( ( cls ` J ) ` x ) )
7	1, 6	sylan 488	. . 3 |- ( ( J e. Comp /\ F e. ( Fil ` X ) ) -> ( J fClus F ) = |^|_ x e. F ( ( cls ` J ) ` x ) )
8		fvex 6201	. . . 4 |- ( ( cls ` J ) ` x ) e. _V
9	8	dfiin3 5381	. . 3 |- |^|_ x e. F ( ( cls ` J ) ` x ) = |^| ran ( x e. F |-> ( ( cls ` J ) ` x ) )
10	7, 9	syl6eq 2672	. 2 |- ( ( J e. Comp /\ F e. ( Fil ` X ) ) -> ( J fClus F ) = |^| ran ( x e. F |-> ( ( cls ` J ) ` x ) ) )
11		simpl 473	. . 3 |- ( ( J e. Comp /\ F e. ( Fil ` X ) ) -> J e. Comp )
12	11	adantr 481	. . . . . . 7 |- ( ( ( J e. Comp /\ F e. ( Fil ` X ) ) /\ x e. F ) -> J e. Comp )
13	12, 1	syl 17	. . . . . 6 |- ( ( ( J e. Comp /\ F e. ( Fil ` X ) ) /\ x e. F ) -> J e. Top )
14		filelss 21656	. . . . . . 7 |- ( ( F e. ( Fil ` X ) /\ x e. F ) -> x C_ X )
15	14	adantll 750	. . . . . 6 |- ( ( ( J e. Comp /\ F e. ( Fil ` X ) ) /\ x e. F ) -> x C_ X )
16	2	clscld 20851	. . . . . 6 |- ( ( J e. Top /\ x C_ X ) -> ( ( cls ` J ) ` x ) e. ( Clsd ` J ) )
17	13, 15, 16	syl2anc 693	. . . . 5 |- ( ( ( J e. Comp /\ F e. ( Fil ` X ) ) /\ x e. F ) -> ( ( cls ` J ) ` x ) e. ( Clsd ` J ) )
18		eqid 2622	. . . . 5 |- ( x e. F |-> ( ( cls ` J ) ` x ) ) = ( x e. F |-> ( ( cls ` J ) ` x ) )
19	17, 18	fmptd 6385	. . . 4 |- ( ( J e. Comp /\ F e. ( Fil ` X ) ) -> ( x e. F |-> ( ( cls ` J ) ` x ) ) : F --> ( Clsd ` J ) )
20		frn 6053	. . . 4 |- ( ( x e. F |-> ( ( cls ` J ) ` x ) ) : F --> ( Clsd ` J ) -> ran ( x e. F |-> ( ( cls ` J ) ` x ) ) C_ ( Clsd ` J ) )
21	19, 20	syl 17	. . 3 |- ( ( J e. Comp /\ F e. ( Fil ` X ) ) -> ran ( x e. F |-> ( ( cls ` J ) ` x ) ) C_ ( Clsd ` J ) )
22		simpr 477	. . . . . 6 |- ( ( J e. Comp /\ F e. ( Fil ` X ) ) -> F e. ( Fil ` X ) )
23	22	adantr 481	. . . . . . . . 9 |- ( ( ( J e. Comp /\ F e. ( Fil ` X ) ) /\ x e. F ) -> F e. ( Fil ` X ) )
24		simpr 477	. . . . . . . . 9 |- ( ( ( J e. Comp /\ F e. ( Fil ` X ) ) /\ x e. F ) -> x e. F )
25	2	clsss3 20863	. . . . . . . . . 10 |- ( ( J e. Top /\ x C_ X ) -> ( ( cls ` J ) ` x ) C_ X )
26	13, 15, 25	syl2anc 693	. . . . . . . . 9 |- ( ( ( J e. Comp /\ F e. ( Fil ` X ) ) /\ x e. F ) -> ( ( cls ` J ) ` x ) C_ X )
27	2	sscls 20860	. . . . . . . . . 10 |- ( ( J e. Top /\ x C_ X ) -> x C_ ( ( cls ` J ) ` x ) )
28	13, 15, 27	syl2anc 693	. . . . . . . . 9 |- ( ( ( J e. Comp /\ F e. ( Fil ` X ) ) /\ x e. F ) -> x C_ ( ( cls ` J ) ` x ) )
29		filss 21657	. . . . . . . . 9 |- ( ( F e. ( Fil ` X ) /\ ( x e. F /\ ( ( cls ` J ) ` x ) C_ X /\ x C_ ( ( cls ` J ) ` x ) ) ) -> ( ( cls ` J ) ` x ) e. F )
30	23, 24, 26, 28, 29	syl13anc 1328	. . . . . . . 8 |- ( ( ( J e. Comp /\ F e. ( Fil ` X ) ) /\ x e. F ) -> ( ( cls ` J ) ` x ) e. F )
31	30, 18	fmptd 6385	. . . . . . 7 |- ( ( J e. Comp /\ F e. ( Fil ` X ) ) -> ( x e. F |-> ( ( cls ` J ) ` x ) ) : F --> F )
32		frn 6053	. . . . . . 7 |- ( ( x e. F |-> ( ( cls ` J ) ` x ) ) : F --> F -> ran ( x e. F |-> ( ( cls ` J ) ` x ) ) C_ F )
33	31, 32	syl 17	. . . . . 6 |- ( ( J e. Comp /\ F e. ( Fil ` X ) ) -> ran ( x e. F |-> ( ( cls ` J ) ` x ) ) C_ F )
34		fiss 8330	. . . . . 6 |- ( ( F e. ( Fil ` X ) /\ ran ( x e. F |-> ( ( cls ` J ) ` x ) ) C_ F ) -> ( fi ` ran ( x e. F |-> ( ( cls ` J ) ` x ) ) ) C_ ( fi ` F ) )
35	22, 33, 34	syl2anc 693	. . . . 5 |- ( ( J e. Comp /\ F e. ( Fil ` X ) ) -> ( fi ` ran ( x e. F |-> ( ( cls ` J ) ` x ) ) ) C_ ( fi ` F ) )
36		filfi 21663	. . . . . 6 |- ( F e. ( Fil ` X ) -> ( fi ` F ) = F )
37	22, 36	syl 17	. . . . 5 |- ( ( J e. Comp /\ F e. ( Fil ` X ) ) -> ( fi ` F ) = F )
38	35, 37	sseqtrd 3641	. . . 4 |- ( ( J e. Comp /\ F e. ( Fil ` X ) ) -> ( fi ` ran ( x e. F |-> ( ( cls ` J ) ` x ) ) ) C_ F )
39		0nelfil 21653	. . . . 5 |- ( F e. ( Fil ` X ) -> -. (/) e. F )
40	22, 39	syl 17	. . . 4 |- ( ( J e. Comp /\ F e. ( Fil ` X ) ) -> -. (/) e. F )
41	38, 40	ssneldd 3606	. . 3 |- ( ( J e. Comp /\ F e. ( Fil ` X ) ) -> -. (/) e. ( fi ` ran ( x e. F |-> ( ( cls ` J ) ` x ) ) ) )
42		cmpfii 21212	. . 3 |- ( ( J e. Comp /\ ran ( x e. F |-> ( ( cls ` J ) ` x ) ) C_ ( Clsd ` J ) /\ -. (/) e. ( fi ` ran ( x e. F |-> ( ( cls ` J ) ` x ) ) ) ) -> |^| ran ( x e. F |-> ( ( cls ` J ) ` x ) ) =/= (/) )
43	11, 21, 41, 42	syl3anc 1326	. 2 |- ( ( J e. Comp /\ F e. ( Fil ` X ) ) -> |^| ran ( x e. F |-> ( ( cls ` J ) ` x ) ) =/= (/) )
44	10, 43	eqnetrd 2861	1 |- ( ( J e. Comp /\ F e. ( Fil ` X ) ) -> ( J fClus F ) =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   ifcif 4086   U.cuni 4436   |^|cint 4475   |^|_ciin 4521   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   ficfi 8316   Topctop 20698   Clsdccld 20820   clsccl 20822   Compccmp 21189   Filcfil 21649   fClus cfcls 21740

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-rep 4771  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-nel 2898  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-fbas 19743  df-top 20699  df-cld 20823  df-cls 20825  df-cmp 21190  df-fil 21650  df-fcls 21745

This theorem is referenced by:  fclscmp  21834  ufilcmp  21836  relcmpcmet  23115