Theorem ufilcmp 21836

Description: A space is compact iff every ultrafilter converges. (Contributed by Jeff Hankins, 11-Dec-2009.) (Proof shortened by Mario Carneiro, 12-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
ufilcmp	|- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( J e. Comp <-> A. f e. ( UFil ` X ) ( J fLim f ) =/= (/) ) )

Distinct variable groups:   f, J   f, X

Proof of Theorem ufilcmp

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ufilfil 21708	. . . . . 6 |- ( f e. ( UFil ` U. J ) -> f e. ( Fil ` U. J ) )
2		eqid 2622	. . . . . . 7 |- U. J = U. J
3	2	fclscmpi 21833	. . . . . 6 |- ( ( J e. Comp /\ f e. ( Fil ` U. J ) ) -> ( J fClus f ) =/= (/) )
4	1, 3	sylan2 491	. . . . 5 |- ( ( J e. Comp /\ f e. ( UFil ` U. J ) ) -> ( J fClus f ) =/= (/) )
5	4	ralrimiva 2966	. . . 4 |- ( J e. Comp -> A. f e. ( UFil ` U. J ) ( J fClus f ) =/= (/) )
6		toponuni 20719	. . . . . . 7 |- ( J e. (TopOn ` X ) -> X = U. J )
7	6	fveq2d 6195	. . . . . 6 |- ( J e. (TopOn ` X ) -> ( UFil ` X ) = ( UFil ` U. J ) )
8	7	raleqdv 3144	. . . . 5 |- ( J e. (TopOn ` X ) -> ( A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) <-> A. f e. ( UFil ` U. J ) ( J fClus f ) =/= (/) ) )
9	8	adantl 482	. . . 4 |- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) <-> A. f e. ( UFil ` U. J ) ( J fClus f ) =/= (/) ) )
10	5, 9	syl5ibr 236	. . 3 |- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( J e. Comp -> A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) ) )
11		ufli 21718	. . . . . . 7 |- ( ( X e. UFL /\ g e. ( Fil ` X ) ) -> E. f e. ( UFil ` X ) g C_ f )
12	11	adantlr 751	. . . . . 6 |- ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) -> E. f e. ( UFil ` X ) g C_ f )
13		r19.29 3072	. . . . . . 7 |- ( ( A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) /\ E. f e. ( UFil ` X ) g C_ f ) -> E. f e. ( UFil ` X ) ( ( J fClus f ) =/= (/) /\ g C_ f ) )
14		simpllr 799	. . . . . . . . . . . . 13 |- ( ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) /\ ( f e. ( UFil ` X ) /\ g C_ f ) ) -> J e. (TopOn ` X ) )
15		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) /\ ( f e. ( UFil ` X ) /\ g C_ f ) ) -> g e. ( Fil ` X ) )
16		simprr 796	. . . . . . . . . . . . 13 |- ( ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) /\ ( f e. ( UFil ` X ) /\ g C_ f ) ) -> g C_ f )
17		fclsss2 21827	. . . . . . . . . . . . 13 |- ( ( J e. (TopOn ` X ) /\ g e. ( Fil ` X ) /\ g C_ f ) -> ( J fClus f ) C_ ( J fClus g ) )
18	14, 15, 16, 17	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) /\ ( f e. ( UFil ` X ) /\ g C_ f ) ) -> ( J fClus f ) C_ ( J fClus g ) )
19		ssn0 3976	. . . . . . . . . . . . 13 |- ( ( ( J fClus f ) C_ ( J fClus g ) /\ ( J fClus f ) =/= (/) ) -> ( J fClus g ) =/= (/) )
20	19	ex 450	. . . . . . . . . . . 12 |- ( ( J fClus f ) C_ ( J fClus g ) -> ( ( J fClus f ) =/= (/) -> ( J fClus g ) =/= (/) ) )
21	18, 20	syl 17	. . . . . . . . . . 11 |- ( ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) /\ ( f e. ( UFil ` X ) /\ g C_ f ) ) -> ( ( J fClus f ) =/= (/) -> ( J fClus g ) =/= (/) ) )
22	21	expr 643	. . . . . . . . . 10 |- ( ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) /\ f e. ( UFil ` X ) ) -> ( g C_ f -> ( ( J fClus f ) =/= (/) -> ( J fClus g ) =/= (/) ) ) )
23	22	com23 86	. . . . . . . . 9 |- ( ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) /\ f e. ( UFil ` X ) ) -> ( ( J fClus f ) =/= (/) -> ( g C_ f -> ( J fClus g ) =/= (/) ) ) )
24	23	impd 447	. . . . . . . 8 |- ( ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) /\ f e. ( UFil ` X ) ) -> ( ( ( J fClus f ) =/= (/) /\ g C_ f ) -> ( J fClus g ) =/= (/) ) )
25	24	rexlimdva 3031	. . . . . . 7 |- ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) -> ( E. f e. ( UFil ` X ) ( ( J fClus f ) =/= (/) /\ g C_ f ) -> ( J fClus g ) =/= (/) ) )
26	13, 25	syl5 34	. . . . . 6 |- ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) -> ( ( A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) /\ E. f e. ( UFil ` X ) g C_ f ) -> ( J fClus g ) =/= (/) ) )
27	12, 26	mpan2d 710	. . . . 5 |- ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) -> ( A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) -> ( J fClus g ) =/= (/) ) )
28	27	ralrimdva 2969	. . . 4 |- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) -> A. g e. ( Fil ` X ) ( J fClus g ) =/= (/) ) )
29		fclscmp 21834	. . . . 5 |- ( J e. (TopOn ` X ) -> ( J e. Comp <-> A. g e. ( Fil ` X ) ( J fClus g ) =/= (/) ) )
30	29	adantl 482	. . . 4 |- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( J e. Comp <-> A. g e. ( Fil ` X ) ( J fClus g ) =/= (/) ) )
31	28, 30	sylibrd 249	. . 3 |- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) -> J e. Comp ) )
32	10, 31	impbid 202	. 2 |- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( J e. Comp <-> A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) ) )
33		uffclsflim 21835	. . . 4 |- ( f e. ( UFil ` X ) -> ( J fClus f ) = ( J fLim f ) )
34	33	neeq1d 2853	. . 3 |- ( f e. ( UFil ` X ) -> ( ( J fClus f ) =/= (/) <-> ( J fLim f ) =/= (/) ) )
35	34	ralbiia 2979	. 2 |- ( A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) <-> A. f e. ( UFil ` X ) ( J fLim f ) =/= (/) )
36	32, 35	syl6bb 276	1 |- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( J e. Comp <-> A. f e. ( UFil ` X ) ( J fLim f ) =/= (/) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   U.cuni 4436   ` cfv 5888  (class class class)co 6650  TopOnctopon 20715   Compccmp 21189   Filcfil 21649   UFilcufil 21703  UFLcufl 21704   fLim cflim 21738   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-fg 19744  df-top 20699  df-topon 20716  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-cmp 21190  df-fil 21650  df-ufil 21705  df-ufl 21706  df-flim 21743  df-fcls 21745

This theorem is referenced by:  alexsub  21849