Theorem flimss1 21777

Description: A limit point of a filter is a limit point in a coarser topology. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Assertion

Ref	Expression
flimss1	|- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) -> ( K fLim F ) C_ ( J fLim F ) )

Proof of Theorem flimss1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . 7 |- U. K = U. K
2	1	flimelbas 21772	. . . . . 6 |- ( x e. ( K fLim F ) -> x e. U. K )
3	2	adantl 482	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> x e. U. K )
4		simpl2 1065	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> F e. ( Fil ` X ) )
5		filunibas 21685	. . . . . . 7 |- ( F e. ( Fil ` X ) -> U. F = X )
6	4, 5	syl 17	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> U. F = X )
7	1	flimfil 21773	. . . . . . . 8 |- ( x e. ( K fLim F ) -> F e. ( Fil ` U. K ) )
8	7	adantl 482	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> F e. ( Fil ` U. K ) )
9		filunibas 21685	. . . . . . 7 |- ( F e. ( Fil ` U. K ) -> U. F = U. K )
10	8, 9	syl 17	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> U. F = U. K )
11	6, 10	eqtr3d 2658	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> X = U. K )
12	3, 11	eleqtrrd 2704	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> x e. X )
13		simpl1 1064	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> J e. (TopOn ` X ) )
14		topontop 20718	. . . . . . 7 |- ( J e. (TopOn ` X ) -> J e. Top )
15	13, 14	syl 17	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> J e. Top )
16		flimtop 21769	. . . . . . 7 |- ( x e. ( K fLim F ) -> K e. Top )
17	16	adantl 482	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> K e. Top )
18		toponuni 20719	. . . . . . . 8 |- ( J e. (TopOn ` X ) -> X = U. J )
19	13, 18	syl 17	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> X = U. J )
20	19, 11	eqtr3d 2658	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> U. J = U. K )
21		simpl3 1066	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> J C_ K )
22		eqid 2622	. . . . . . 7 |- U. J = U. J
23	22, 1	topssnei 20928	. . . . . 6 |- ( ( ( J e. Top /\ K e. Top /\ U. J = U. K ) /\ J C_ K ) -> ( ( nei ` J ) ` { x } ) C_ ( ( nei ` K ) ` { x } ) )
24	15, 17, 20, 21, 23	syl31anc 1329	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> ( ( nei ` J ) ` { x } ) C_ ( ( nei ` K ) ` { x } ) )
25		flimneiss 21770	. . . . . 6 |- ( x e. ( K fLim F ) -> ( ( nei ` K ) ` { x } ) C_ F )
26	25	adantl 482	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> ( ( nei ` K ) ` { x } ) C_ F )
27	24, 26	sstrd 3613	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> ( ( nei ` J ) ` { x } ) C_ F )
28		elflim 21775	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) ) -> ( x e. ( J fLim F ) <-> ( x e. X /\ ( ( nei ` J ) ` { x } ) C_ F ) ) )
29	13, 4, 28	syl2anc 693	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> ( x e. ( J fLim F ) <-> ( x e. X /\ ( ( nei ` J ) ` { x } ) C_ F ) ) )
30	12, 27, 29	mpbir2and 957	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) /\ x e. ( K fLim F ) ) -> x e. ( J fLim F ) )
31	30	ex 450	. 2 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) -> ( x e. ( K fLim F ) -> x e. ( J fLim F ) ) )
32	31	ssrdv 3609	1 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) /\ J C_ K ) -> ( K fLim F ) C_ ( J fLim F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   {csn 4177   U.cuni 4436   ` cfv 5888  (class class class)co 6650   Topctop 20698  TopOnctopon 20715   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-iun 4522  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-ntr 20824  df-nei 20902  df-fil 21650  df-flim 21743

This theorem is referenced by:  flimcf  21786