Theorem ufilcmp 20618

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 20490	. . . . . 6 |- ( f e. ( UFil ` U. J ) -> f e. ( Fil ` U. J ) )
2		eqid 2382	. . . . . . 7 |- U. J = U. J
3	2	fclscmpi 20615	. . . . . 6 |- ( ( J e. Comp /\ f e. ( Fil ` U. J ) ) -> ( J fClus f ) =/= (/) )
4	1, 3	sylan2 472	. . . . 5 |- ( ( J e. Comp /\ f e. ( UFil ` U. J ) ) -> ( J fClus f ) =/= (/) )
5	4	ralrimiva 2796	. . . 4 |- ( J e. Comp -> A. f e. ( UFil ` U. J ) ( J fClus f ) =/= (/) )
6		toponuni 19513	. . . . . . 7 |- ( J e. (TopOn ` X ) -> X = U. J )
7	6	fveq2d 5778	. . . . . 6 |- ( J e. (TopOn ` X ) -> ( UFil ` X ) = ( UFil ` U. J ) )
8	7	raleqdv 2985	. . . . 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 464	. . . 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 20500	. . . . . . 7 |- ( ( X e. UFL /\ g e. ( Fil ` X ) ) -> E. f e. ( UFil ` X ) g C_ f )
12	11	adantlr 712	. . . . . 6 |- ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) -> E. f e. ( UFil ` X ) g C_ f )
13		r19.29 2917	. . . . . . 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 753	. . . . . . . . . . . . 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 755	. . . . . . . . . . . . 13 |- ( ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) /\ ( f e. ( UFil ` X ) /\ g C_ f ) ) -> g C_ f )
17		fclsss2 20609	. . . . . . . . . . . . 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 1226	. . . . . . . . . . . 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 3745	. . . . . . . . . . . . 13 |- ( ( ( J fClus f ) C_ ( J fClus g ) /\ ( J fClus f ) =/= (/) ) -> ( J fClus g ) =/= (/) )
20	19	ex 432	. . . . . . . . . . . 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 613	. . . . . . . . . 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 429	. . . . . . . 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 2874	. . . . . . 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 672	. . . . 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 2800	. . . 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 20616	. . . . 5 |- ( J e. (TopOn ` X ) -> ( J e. Comp <-> A. g e. ( Fil ` X ) ( J fClus g ) =/= (/) ) )
30	29	adantl 464	. . . 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 20617	. . . 4 |- ( f e. ( UFil ` X ) -> ( J fClus f ) = ( J fLim f ) )
34	33	neeq1d 2659	. . 3 |- ( f e. ( UFil ` X ) -> ( ( J fClus f ) =/= (/) <-> ( J fLim f ) =/= (/) ) )
35	34	ralbiia 2812	. 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 367   e. wcel 1826   =/= wne 2577   A.wral 2732   E.wrex 2733   C_ wss 3389   (/)c0 3711   U.cuni 4163   ` cfv 5496  (class class class)co 6196  TopOnctopon 19480   Compccmp 19972   Filcfil 20431   UFilcufil 20485  UFLcufl 20486   fLim cflim 20520   fClus cfcls 20522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fi 7786  df-fbas 18529  df-fg 18530  df-top 19484  df-topon 19487  df-cld 19605  df-ntr 19606  df-cls 19607  df-nei 19685  df-cmp 19973  df-fil 20432  df-ufil 20487  df-ufl 20488  df-flim 20525  df-fcls 20527

This theorem is referenced by:  alexsub  20630