Theorem ufilcmp 21059

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 20931	. . . . . 6 |- ( f e. ( UFil ` U. J ) -> f e. ( Fil ` U. J ) )
2		eqid 2453	. . . . . . 7 |- U. J = U. J
3	2	fclscmpi 21056	. . . . . 6 |- ( ( J e. Comp /\ f e. ( Fil ` U. J ) ) -> ( J fClus f ) =/= (/) )
4	1, 3	sylan2 477	. . . . 5 |- ( ( J e. Comp /\ f e. ( UFil ` U. J ) ) -> ( J fClus f ) =/= (/) )
5	4	ralrimiva 2804	. . . 4 |- ( J e. Comp -> A. f e. ( UFil ` U. J ) ( J fClus f ) =/= (/) )
6		toponuni 19954	. . . . . . 7 |- ( J e. (TopOn ` X ) -> X = U. J )
7	6	fveq2d 5874	. . . . . 6 |- ( J e. (TopOn ` X ) -> ( UFil ` X ) = ( UFil ` U. J ) )
8	7	raleqdv 2995	. . . . 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 468	. . . 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 225	. . 3 |- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( J e. Comp -> A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) ) )
11		ufli 20941	. . . . . . 7 |- ( ( X e. UFL /\ g e. ( Fil ` X ) ) -> E. f e. ( UFil ` X ) g C_ f )
12	11	adantlr 722	. . . . . 6 |- ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) -> E. f e. ( UFil ` X ) g C_ f )
13		r19.29 2927	. . . . . . 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 770	. . . . . . . . . . . . 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 763	. . . . . . . . . . . . 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 767	. . . . . . . . . . . . 13 |- ( ( ( ( X e. UFL /\ J e. (TopOn ` X ) ) /\ g e. ( Fil ` X ) ) /\ ( f e. ( UFil ` X ) /\ g C_ f ) ) -> g C_ f )
17		fclsss2 21050	. . . . . . . . . . . . 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 1269	. . . . . . . . . . . 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 3769	. . . . . . . . . . . . 13 |- ( ( ( J fClus f ) C_ ( J fClus g ) /\ ( J fClus f ) =/= (/) ) -> ( J fClus g ) =/= (/) )
20	19	ex 436	. . . . . . . . . . . 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 620	. . . . . . . . . 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 81	. . . . . . . . 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 433	. . . . . . . 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 2881	. . . . . . 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 33	. . . . . 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 681	. . . . 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 2808	. . . 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 21057	. . . . 5 |- ( J e. (TopOn ` X ) -> ( J e. Comp <-> A. g e. ( Fil ` X ) ( J fClus g ) =/= (/) ) )
30	29	adantl 468	. . . 4 |- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( J e. Comp <-> A. g e. ( Fil ` X ) ( J fClus g ) =/= (/) ) )
31	28, 30	sylibrd 238	. . 3 |- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) -> J e. Comp ) )
32	10, 31	impbid 194	. 2 |- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( J e. Comp <-> A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) ) )
33		uffclsflim 21058	. . . 4 |- ( f e. ( UFil ` X ) -> ( J fClus f ) = ( J fLim f ) )
34	33	neeq1d 2685	. . 3 |- ( f e. ( UFil ` X ) -> ( ( J fClus f ) =/= (/) <-> ( J fLim f ) =/= (/) ) )
35	34	ralbiia 2820	. 2 |- ( A. f e. ( UFil ` X ) ( J fClus f ) =/= (/) <-> A. f e. ( UFil ` X ) ( J fLim f ) =/= (/) )
36	32, 35	syl6bb 265	1 |- ( ( X e. UFL /\ J e. (TopOn ` X ) ) -> ( J e. Comp <-> A. f e. ( UFil ` X ) ( J fLim f ) =/= (/) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   e. wcel 1889   =/= wne 2624   A.wral 2739   E.wrex 2740   C_ wss 3406   (/)c0 3733   U.cuni 4201   ` cfv 5585  (class class class)co 6295  TopOnctopon 19930   Compccmp 20413   Filcfil 20872   UFilcufil 20926  UFLcufl 20927   fLim cflim 20961   fClus cfcls 20963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fi 7930  df-fbas 18979  df-fg 18980  df-top 19933  df-topon 19935  df-cld 20046  df-ntr 20047  df-cls 20048  df-nei 20126  df-cmp 20414  df-fil 20873  df-ufil 20928  df-ufl 20929  df-flim 20966  df-fcls 20968

This theorem is referenced by:  alexsub  21072