Theorem fclsss2 21827

Description: A finer filter has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
fclsss2	|- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ F C_ G ) -> ( J fClus G ) C_ ( J fClus F ) )

Proof of Theorem fclsss2

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

Step	Hyp	Ref	Expression
1		simpl3 1066	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ F C_ G ) /\ x e. ( J fClus G ) ) -> F C_ G )
2		ssralv 3666	. . . . . 6 |- ( F C_ G -> ( A. s e. G x e. ( ( cls ` J ) ` s ) -> A. s e. F x e. ( ( cls ` J ) ` s ) ) )
3	1, 2	syl 17	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ F C_ G ) /\ x e. ( J fClus G ) ) -> ( A. s e. G x e. ( ( cls ` J ) ` s ) -> A. s e. F x e. ( ( cls ` J ) ` s ) ) )
4		simpl1 1064	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ F C_ G ) /\ x e. ( J fClus G ) ) -> J e. (TopOn ` X ) )
5		fclstopon 21816	. . . . . . . 8 |- ( x e. ( J fClus G ) -> ( J e. (TopOn ` X ) <-> G e. ( Fil ` X ) ) )
6	5	adantl 482	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ F C_ G ) /\ x e. ( J fClus G ) ) -> ( J e. (TopOn ` X ) <-> G e. ( Fil ` X ) ) )
7	4, 6	mpbid 222	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ F C_ G ) /\ x e. ( J fClus G ) ) -> G e. ( Fil ` X ) )
8		isfcls2 21817	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ G e. ( Fil ` X ) ) -> ( x e. ( J fClus G ) <-> A. s e. G x e. ( ( cls ` J ) ` s ) ) )
9	4, 7, 8	syl2anc 693	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ F C_ G ) /\ x e. ( J fClus G ) ) -> ( x e. ( J fClus G ) <-> A. s e. G x e. ( ( cls ` J ) ` s ) ) )
10		simpl2 1065	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ F C_ G ) /\ x e. ( J fClus G ) ) -> F e. ( Fil ` X ) )
11		isfcls2 21817	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) ) -> ( x e. ( J fClus F ) <-> A. s e. F x e. ( ( cls ` J ) ` s ) ) )
12	4, 10, 11	syl2anc 693	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ F C_ G ) /\ x e. ( J fClus G ) ) -> ( x e. ( J fClus F ) <-> A. s e. F x e. ( ( cls ` J ) ` s ) ) )
13	3, 9, 12	3imtr4d 283	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ F C_ G ) /\ x e. ( J fClus G ) ) -> ( x e. ( J fClus G ) -> x e. ( J fClus F ) ) )
14	13	ex 450	. . 3 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ F C_ G ) -> ( x e. ( J fClus G ) -> ( x e. ( J fClus G ) -> x e. ( J fClus F ) ) ) )
15	14	pm2.43d 53	. 2 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ F C_ G ) -> ( x e. ( J fClus G ) -> x e. ( J fClus F ) ) )
16	15	ssrdv 3609	1 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ F C_ G ) -> ( J fClus G ) C_ ( J fClus F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   A.wral 2912   C_ wss 3574   ` cfv 5888  (class class class)co 6650  TopOnctopon 20715   clsccl 20822   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-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-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-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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fbas 19743  df-topon 20716  df-fil 21650  df-fcls 21745

This theorem is referenced by:  fclsfnflim  21831  ufilcmp  21836  cnpfcfi  21844