Theorem ufilcmp 19732

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 19604	. . . . . 6 |- ( f e. ( UFil ` U. J ) -> f e. ( Fil ` U. J ) )
2		eqid 2452	. . . . . . 7 |- U. J = U. J
3	2	fclscmpi 19729	. . . . . 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 2827	. . . 4 |- ( J e. Comp -> A. f e. ( UFil ` U. J ) ( J fClus f ) =/= (/) )
6		toponuni 18659	. . . . . . 7 |- ( J e. (TopOn ` X ) -> X = U. J )
7	6	fveq2d 5798	. . . . . 6 |- ( J e. (TopOn ` X ) -> ( UFil ` X ) = ( UFil ` U. J ) )
8	7	raleqdv 3023	. . . . 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 19614	. . . . . . 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 2957	. . . . . . 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 19723	. . . . . . . . . . . . 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 1219	. . . . . . . . . . . 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 3773	. . . . . . . . . . . . 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 2941	. . . . . . 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 2906	. . . 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 19730	. . . . 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 19731	. . . 4 |- ( f e. ( UFil ` X ) -> ( J fClus f ) = ( J fLim f ) )
34	33	neeq1d 2726	. . 3 |- ( f e. ( UFil ` X ) -> ( ( J fClus f ) =/= (/) <-> ( J fLim f ) =/= (/) ) )
35	34	ralbiia 2835	. 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 1758   =/= wne 2645   A.wral 2796   E.wrex 2797   C_ wss 3431   (/)c0 3740   U.cuni 4194   ` cfv 5521  (class class class)co 6195  TopOnctopon 18626   Compccmp 19116   Filcfil 19545   UFilcufil 19599  UFLcufl 19600   fLim cflim 19634   fClus cfcls 19636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-fi 7767  df-fbas 17934  df-fg 17935  df-top 18630  df-topon 18633  df-cld 18750  df-ntr 18751  df-cls 18752  df-nei 18829  df-cmp 19117  df-fil 19546  df-ufil 19601  df-ufl 19602  df-flim 19639  df-fcls 19641

This theorem is referenced by:  alexsub  19744