Theorem flimfil 21773

Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
flimuni.1	|- X = U. J

Assertion

Ref	Expression
flimfil	|- ( A e. ( J fLim F ) -> F e. ( Fil ` X ) )

Proof of Theorem flimfil

Step	Hyp	Ref	Expression
1		flimuni.1	. . . . . 6 |- X = U. J
2	1	elflim2 21768	. . . . 5 |- ( A e. ( J fLim F ) <-> ( ( J e. Top /\ F e. U. ran Fil /\ F C_ ~P X ) /\ ( A e. X /\ ( ( nei ` J ) ` { A } ) C_ F ) ) )
3	2	simplbi 476	. . . 4 |- ( A e. ( J fLim F ) -> ( J e. Top /\ F e. U. ran Fil /\ F C_ ~P X ) )
4	3	simp2d 1074	. . 3 |- ( A e. ( J fLim F ) -> F e. U. ran Fil )
5		filunirn 21686	. . 3 |- ( F e. U. ran Fil <-> F e. ( Fil ` U. F ) )
6	4, 5	sylib 208	. 2 |- ( A e. ( J fLim F ) -> F e. ( Fil ` U. F ) )
7	3	simp3d 1075	. . . . 5 |- ( A e. ( J fLim F ) -> F C_ ~P X )
8		sspwuni 4611	. . . . 5 |- ( F C_ ~P X <-> U. F C_ X )
9	7, 8	sylib 208	. . . 4 |- ( A e. ( J fLim F ) -> U. F C_ X )
10		flimneiss 21770	. . . . . 6 |- ( A e. ( J fLim F ) -> ( ( nei ` J ) ` { A } ) C_ F )
11		flimtop 21769	. . . . . . 7 |- ( A e. ( J fLim F ) -> J e. Top )
12	1	topopn 20711	. . . . . . . 8 |- ( J e. Top -> X e. J )
13	11, 12	syl 17	. . . . . . 7 |- ( A e. ( J fLim F ) -> X e. J )
14	1	flimelbas 21772	. . . . . . 7 |- ( A e. ( J fLim F ) -> A e. X )
15		opnneip 20923	. . . . . . 7 |- ( ( J e. Top /\ X e. J /\ A e. X ) -> X e. ( ( nei ` J ) ` { A } ) )
16	11, 13, 14, 15	syl3anc 1326	. . . . . 6 |- ( A e. ( J fLim F ) -> X e. ( ( nei ` J ) ` { A } ) )
17	10, 16	sseldd 3604	. . . . 5 |- ( A e. ( J fLim F ) -> X e. F )
18		elssuni 4467	. . . . 5 |- ( X e. F -> X C_ U. F )
19	17, 18	syl 17	. . . 4 |- ( A e. ( J fLim F ) -> X C_ U. F )
20	9, 19	eqssd 3620	. . 3 |- ( A e. ( J fLim F ) -> U. F = X )
21	20	fveq2d 6195	. 2 |- ( A e. ( J fLim F ) -> ( Fil ` U. F ) = ( Fil ` X ) )
22	6, 21	eleqtrd 2703	1 |- ( A e. ( J fLim F ) -> F e. ( Fil ` X ) )

Syntax hints:   -> wi 4   /\ 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   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

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-nei 20902  df-fil 21650  df-flim 21743

This theorem is referenced by:  flimtopon  21774  flimss1  21777  flimclsi  21782  hausflimlem  21783  flimsncls  21790  cnpflfi  21803  flimfcls  21830  flimcfil  23112