Theorem fclsopni 21819

Description: An open neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Assertion

Ref	Expression
fclsopni	|- ( ( A e. ( J fClus F ) /\ ( U e. J /\ A e. U /\ S e. F ) ) -> ( U i^i S ) =/= (/) )

Proof of Theorem fclsopni

Dummy variables o s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . . . 9 |- U. J = U. J
2	1	fclsfil 21814	. . . . . . . 8 |- ( A e. ( J fClus F ) -> F e. ( Fil ` U. J ) )
3		fclstopon 21816	. . . . . . . 8 |- ( A e. ( J fClus F ) -> ( J e. (TopOn ` U. J ) <-> F e. ( Fil ` U. J ) ) )
4	2, 3	mpbird 247	. . . . . . 7 |- ( A e. ( J fClus F ) -> J e. (TopOn ` U. J ) )
5		fclsopn 21818	. . . . . . 7 |- ( ( J e. (TopOn ` U. J ) /\ F e. ( Fil ` U. J ) ) -> ( A e. ( J fClus F ) <-> ( A e. U. J /\ A. o e. J ( A e. o -> A. s e. F ( o i^i s ) =/= (/) ) ) ) )
6	4, 2, 5	syl2anc 693	. . . . . 6 |- ( A e. ( J fClus F ) -> ( A e. ( J fClus F ) <-> ( A e. U. J /\ A. o e. J ( A e. o -> A. s e. F ( o i^i s ) =/= (/) ) ) ) )
7	6	ibi 256	. . . . 5 |- ( A e. ( J fClus F ) -> ( A e. U. J /\ A. o e. J ( A e. o -> A. s e. F ( o i^i s ) =/= (/) ) ) )
8	7	simprd 479	. . . 4 |- ( A e. ( J fClus F ) -> A. o e. J ( A e. o -> A. s e. F ( o i^i s ) =/= (/) ) )
9		eleq2 2690	. . . . . 6 |- ( o = U -> ( A e. o <-> A e. U ) )
10		ineq1 3807	. . . . . . . 8 |- ( o = U -> ( o i^i s ) = ( U i^i s ) )
11	10	neeq1d 2853	. . . . . . 7 |- ( o = U -> ( ( o i^i s ) =/= (/) <-> ( U i^i s ) =/= (/) ) )
12	11	ralbidv 2986	. . . . . 6 |- ( o = U -> ( A. s e. F ( o i^i s ) =/= (/) <-> A. s e. F ( U i^i s ) =/= (/) ) )
13	9, 12	imbi12d 334	. . . . 5 |- ( o = U -> ( ( A e. o -> A. s e. F ( o i^i s ) =/= (/) ) <-> ( A e. U -> A. s e. F ( U i^i s ) =/= (/) ) ) )
14	13	rspccv 3306	. . . 4 |- ( A. o e. J ( A e. o -> A. s e. F ( o i^i s ) =/= (/) ) -> ( U e. J -> ( A e. U -> A. s e. F ( U i^i s ) =/= (/) ) ) )
15	8, 14	syl 17	. . 3 |- ( A e. ( J fClus F ) -> ( U e. J -> ( A e. U -> A. s e. F ( U i^i s ) =/= (/) ) ) )
16		ineq2 3808	. . . . 5 |- ( s = S -> ( U i^i s ) = ( U i^i S ) )
17	16	neeq1d 2853	. . . 4 |- ( s = S -> ( ( U i^i s ) =/= (/) <-> ( U i^i S ) =/= (/) ) )
18	17	rspccv 3306	. . 3 |- ( A. s e. F ( U i^i s ) =/= (/) -> ( S e. F -> ( U i^i S ) =/= (/) ) )
19	15, 18	syl8 76	. 2 |- ( A e. ( J fClus F ) -> ( U e. J -> ( A e. U -> ( S e. F -> ( U i^i S ) =/= (/) ) ) ) )
20	19	3imp2 1282	1 |- ( ( A e. ( J fClus F ) /\ ( U e. J /\ A e. U /\ S e. F ) ) -> ( U i^i S ) =/= (/) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   i^i cin 3573   (/)c0 3915   U.cuni 4436   ` cfv 5888  (class class class)co 6650  TopOnctopon 20715   Filcfil 21649   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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  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-id 5024  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-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-fbas 19743  df-top 20699  df-topon 20716  df-cld 20823  df-ntr 20824  df-cls 20825  df-fil 21650  df-fcls 21745

This theorem is referenced by:  fclsneii  21821  supnfcls  21824  flimfnfcls  21832  cfilfcls  23072