Theorem flimclsi 21782

Description: The convergent points of a filter are a subset of the closure of any of the filter sets. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Assertion

Ref	Expression
flimclsi	|- ( S e. F -> ( J fLim F ) C_ ( ( cls ` J ) ` S ) )

Proof of Theorem flimclsi

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . . 8 |- U. J = U. J
2	1	flimfil 21773	. . . . . . 7 |- ( x e. ( J fLim F ) -> F e. ( Fil ` U. J ) )
3	2	ad2antlr 763	. . . . . 6 |- ( ( ( S e. F /\ x e. ( J fLim F ) ) /\ y e. ( ( nei ` J ) ` { x } ) ) -> F e. ( Fil ` U. J ) )
4		flimnei 21771	. . . . . . 7 |- ( ( x e. ( J fLim F ) /\ y e. ( ( nei ` J ) ` { x } ) ) -> y e. F )
5	4	adantll 750	. . . . . 6 |- ( ( ( S e. F /\ x e. ( J fLim F ) ) /\ y e. ( ( nei ` J ) ` { x } ) ) -> y e. F )
6		simpll 790	. . . . . 6 |- ( ( ( S e. F /\ x e. ( J fLim F ) ) /\ y e. ( ( nei ` J ) ` { x } ) ) -> S e. F )
7		filinn0 21664	. . . . . 6 |- ( ( F e. ( Fil ` U. J ) /\ y e. F /\ S e. F ) -> ( y i^i S ) =/= (/) )
8	3, 5, 6, 7	syl3anc 1326	. . . . 5 |- ( ( ( S e. F /\ x e. ( J fLim F ) ) /\ y e. ( ( nei ` J ) ` { x } ) ) -> ( y i^i S ) =/= (/) )
9	8	ralrimiva 2966	. . . 4 |- ( ( S e. F /\ x e. ( J fLim F ) ) -> A. y e. ( ( nei ` J ) ` { x } ) ( y i^i S ) =/= (/) )
10		flimtop 21769	. . . . . 6 |- ( x e. ( J fLim F ) -> J e. Top )
11	10	adantl 482	. . . . 5 |- ( ( S e. F /\ x e. ( J fLim F ) ) -> J e. Top )
12		filelss 21656	. . . . . . 7 |- ( ( F e. ( Fil ` U. J ) /\ S e. F ) -> S C_ U. J )
13	12	ancoms 469	. . . . . 6 |- ( ( S e. F /\ F e. ( Fil ` U. J ) ) -> S C_ U. J )
14	2, 13	sylan2 491	. . . . 5 |- ( ( S e. F /\ x e. ( J fLim F ) ) -> S C_ U. J )
15	1	flimelbas 21772	. . . . . 6 |- ( x e. ( J fLim F ) -> x e. U. J )
16	15	adantl 482	. . . . 5 |- ( ( S e. F /\ x e. ( J fLim F ) ) -> x e. U. J )
17	1	neindisj2 20927	. . . . 5 |- ( ( J e. Top /\ S C_ U. J /\ x e. U. J ) -> ( x e. ( ( cls ` J ) ` S ) <-> A. y e. ( ( nei ` J ) ` { x } ) ( y i^i S ) =/= (/) ) )
18	11, 14, 16, 17	syl3anc 1326	. . . 4 |- ( ( S e. F /\ x e. ( J fLim F ) ) -> ( x e. ( ( cls ` J ) ` S ) <-> A. y e. ( ( nei ` J ) ` { x } ) ( y i^i S ) =/= (/) ) )
19	9, 18	mpbird 247	. . 3 |- ( ( S e. F /\ x e. ( J fLim F ) ) -> x e. ( ( cls ` J ) ` S ) )
20	19	ex 450	. 2 |- ( S e. F -> ( x e. ( J fLim F ) -> x e. ( ( cls ` J ) ` S ) ) )
21	20	ssrdv 3609	1 |- ( S e. F -> ( J fLim F ) C_ ( ( cls ` J ) ` S ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   =/= wne 2794   A.wral 2912   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   U.cuni 4436   ` cfv 5888  (class class class)co 6650   Topctop 20698   clsccl 20822   neicnei 20901   Filcfil 21649   fLim cflim 21738

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-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-fil 21650  df-flim 21743

This theorem is referenced by:  flimcls  21789  flimfcls  21830  cnextcn  21871  cmetss  23113  minveclem4  23203