Theorem ufilcmp 20399

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 20271	. . . . . 6 |- ( f e. ( UFil ` U. J ) -> f e. ( Fil ` U. J ) )
2		eqid 2441	. . . . . . 7 |- U. J = U. J
3	2	fclscmpi 20396	. . . . . 6 |- ( ( J e. Comp /\ f e. ( Fil ` U. J ) ) -> ( J fClus f ) =/= (/) )
4	1, 3	sylan2 474	. . . . 5 |- ( ( J e. Comp /\ f e. ( UFil ` U. J ) ) -> ( J fClus f ) =/= (/) )
5	4	ralrimiva 2855	. . . 4 |- ( J e. Comp -> A. f e. ( UFil ` U. J ) ( J fClus f ) =/= (/) )
6		toponuni 19295	. . . . . . 7 |- ( J e. (TopOn ` X ) -> X = U. J )
7	6	fveq2d 5856	. . . . . 6 |- ( J e. (TopOn ` X ) -> ( UFil ` X ) = ( UFil ` U. J ) )
8	7	raleqdv 3044	. . . . 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 466	. . . 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 221	. . 3 |- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( J e. Comp -> A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) ) )
11		ufli 20281	. . . . . . 7 |- ( ( X e. UFL /\ g e. ( Fil ` X ) ) -> E. f e. ( UFil ` X ) g C_ f )
12	11	adantlr 714	. . . . . 6 |- ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) -> E. f e. ( UFil ` X ) g C_ f )
13		r19.29 2976	. . . . . . 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 758	. . . . . . . . . . . . 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 754	. . . . . . . . . . . . 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 756	. . . . . . . . . . . . 13 |- ( ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) /\ ( f e. ( UFil ` X ) /\ g C_ f ) ) -> g C_ f )
17		fclsss2 20390	. . . . . . . . . . . . 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 1227	. . . . . . . . . . . 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 3800	. . . . . . . . . . . . 13 |- ( ( ( J fClus f ) C_ ( J fClus g ) /\ ( J fClus f ) =/= (/) ) -> ( J fClus g ) =/= (/) )
20	19	ex 434	. . . . . . . . . . . 12 |- ( ( J fClus f ) C_ ( J fClus g ) -> ( ( J fClus f ) =/= (/) -> ( J fClus g ) =/= (/) ) )
21	18, 20	syl 16	. . . . . . . . . . 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 615	. . . . . . . . . 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 78	. . . . . . . . 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 431	. . . . . . . 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 2933	. . . . . . 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 32	. . . . . 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 674	. . . . 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 2859	. . . 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 20397	. . . . 5 |- ( J e. (TopOn ` X ) -> ( J e. Comp <-> A. g e. ( Fil ` X ) ( J fClus g ) =/= (/) ) )
30	29	adantl 466	. . . 4 |- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( J e. Comp <-> A. g e. ( Fil ` X ) ( J fClus g ) =/= (/) ) )
31	28, 30	sylibrd 234	. . 3 |- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) -> J e. Comp ) )
32	10, 31	impbid 191	. 2 |- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( J e. Comp <-> A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) ) )
33		uffclsflim 20398	. . . 4 |- ( f e. ( UFil ` X ) -> ( J fClus f ) = ( J fLim f ) )
34	33	neeq1d 2718	. . 3 |- ( f e. ( UFil ` X ) -> ( ( J fClus f ) =/= (/) <-> ( J fLim f ) =/= (/) ) )
35	34	ralbiia 2871	. 2 |- ( A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) <-> A. f e. ( UFil ` X ) ( J fLim f ) =/= (/) )
36	32, 35	syl6bb 261	1 |- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( J e. Comp <-> A. f e. ( UFil ` X ) ( J fLim f ) =/= (/) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   e. wcel 1802   =/= wne 2636   A.wral 2791   E.wrex 2792   C_ wss 3458   (/)c0 3767   U.cuni 4230   ` cfv 5574  (class class class)co 6277  TopOnctopon 19262   Compccmp 19752   Filcfil 20212   UFilcufil 20266  UFLcufl 20267   fLim cflim 20301   fClus cfcls 20303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fi 7869  df-fbas 18284  df-fg 18285  df-top 19266  df-topon 19269  df-cld 19386  df-ntr 19387  df-cls 19388  df-nei 19465  df-cmp 19753  df-fil 20213  df-ufil 20268  df-ufl 20269  df-flim 20306  df-fcls 20308

This theorem is referenced by:  alexsub  20411