Theorem elflim 21775

Description: The predicate "is a limit point of a filter." (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
elflim	|- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) ) -> ( A e. ( J fLim F ) <-> ( A e. X /\ ( ( nei ` J ) ` { A } ) C_ F ) ) )

Proof of Theorem elflim

Step	Hyp	Ref	Expression
1		topontop 20718	. . . 4 |- ( J e. (TopOn ` X ) -> J e. Top )
2	1	adantr 481	. . 3 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) ) -> J e. Top )
3		fvssunirn 6217	. . . . 5 |- ( Fil ` X ) C_ U. ran Fil
4	3	sseli 3599	. . . 4 |- ( F e. ( Fil ` X ) -> F e. U. ran Fil )
5	4	adantl 482	. . 3 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) ) -> F e. U. ran Fil )
6		filsspw 21655	. . . . 5 |- ( F e. ( Fil ` X ) -> F C_ ~P X )
7	6	adantl 482	. . . 4 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) ) -> F C_ ~P X )
8		toponuni 20719	. . . . . 6 |- ( J e. (TopOn ` X ) -> X = U. J )
9	8	adantr 481	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) ) -> X = U. J )
10	9	pweqd 4163	. . . 4 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) ) -> ~P X = ~P U. J )
11	7, 10	sseqtrd 3641	. . 3 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) ) -> F C_ ~P U. J )
12		eqid 2622	. . . . 5 |- U. J = U. J
13	12	elflim2 21768	. . . 4 |- ( A e. ( J fLim F ) <-> ( ( J e. Top /\ F e. U. ran Fil /\ F C_ ~P U. J ) /\ ( A e. U. J /\ ( ( nei ` J ) ` { A } ) C_ F ) ) )
14	13	baib 944	. . 3 |- ( ( J e. Top /\ F e. U. ran Fil /\ F C_ ~P U. J ) -> ( A e. ( J fLim F ) <-> ( A e. U. J /\ ( ( nei ` J ) ` { A } ) C_ F ) ) )
15	2, 5, 11, 14	syl3anc 1326	. 2 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) ) -> ( A e. ( J fLim F ) <-> ( A e. U. J /\ ( ( nei ` J ) ` { A } ) C_ F ) ) )
16	9	eleq2d 2687	. . 3 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) ) -> ( A e. X <-> A e. U. J ) )
17	16	anbi1d 741	. 2 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) ) -> ( ( A e. X /\ ( ( nei ` J ) ` { A } ) C_ F ) <-> ( A e. U. J /\ ( ( nei ` J ) ` { A } ) C_ F ) ) )
18	15, 17	bitr4d 271	1 |- ( ( J e. (TopOn ` X ) /\ F e. ( Fil ` X ) ) -> ( A e. ( J fLim F ) <-> ( A e. X /\ ( ( nei ` J ) ` { A } ) C_ F ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   {csn 4177   U.cuni 4436   ran crn 5115   ` 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-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-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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fbas 19743  df-top 20699  df-topon 20716  df-fil 21650  df-flim 21743

This theorem is referenced by:  flimss2  21776  flimss1  21777  neiflim  21778  flimopn  21779  hausflim  21785  flimclslem  21788  flfnei  21795  fclsfnflim  21831