Theorem flfnei 21795

Description: The property of being a limit point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Assertion

Ref	Expression
flfnei	|- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( A e. ( ( J fLimf L ) ` F ) <-> ( A e. X /\ A. n e. ( ( nei ` J ) ` { A } ) E. s e. L ( F " s ) C_ n ) ) )

Distinct variable groups:   n, s, F   A, n   n, J, s   n, L, s   n, X, s   n, Y, s

Allowed substitution hint:   A( s)

Proof of Theorem flfnei

Step	Hyp	Ref	Expression
1		flfval 21794	. . 3 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( ( J fLimf L ) ` F ) = ( J fLim ( ( X FilMap F ) ` L ) ) )
2	1	eleq2d 2687	. 2 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( A e. ( ( J fLimf L ) ` F ) <-> A e. ( J fLim ( ( X FilMap F ) ` L ) ) ) )
3		simp1 1061	. . 3 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> J e. (TopOn ` X ) )
4		toponmax 20730	. . . . 5 |- ( J e. (TopOn ` X ) -> X e. J )
5	4	3ad2ant1 1082	. . . 4 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> X e. J )
6		filfbas 21652	. . . . 5 |- ( L e. ( Fil ` Y ) -> L e. ( fBas ` Y ) )
7	6	3ad2ant2 1083	. . . 4 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> L e. ( fBas ` Y ) )
8		simp3 1063	. . . 4 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> F : Y --> X )
9		fmfil 21748	. . . 4 |- ( ( X e. J /\ L e. ( fBas ` Y ) /\ F : Y --> X ) -> ( ( X FilMap F ) ` L ) e. ( Fil ` X ) )
10	5, 7, 8, 9	syl3anc 1326	. . 3 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( ( X FilMap F ) ` L ) e. ( Fil ` X ) )
11		elflim 21775	. . 3 |- ( ( J e. (TopOn ` X ) /\ ( ( X FilMap F ) ` L ) e. ( Fil ` X ) ) -> ( A e. ( J fLim ( ( X FilMap F ) ` L ) ) <-> ( A e. X /\ ( ( nei ` J ) ` { A } ) C_ ( ( X FilMap F ) ` L ) ) ) )
12	3, 10, 11	syl2anc 693	. 2 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( A e. ( J fLim ( ( X FilMap F ) ` L ) ) <-> ( A e. X /\ ( ( nei ` J ) ` { A } ) C_ ( ( X FilMap F ) ` L ) ) ) )
13		dfss3 3592	. . . 4 |- ( ( ( nei ` J ) ` { A } ) C_ ( ( X FilMap F ) ` L ) <-> A. n e. ( ( nei ` J ) ` { A } ) n e. ( ( X FilMap F ) ` L ) )
14		topontop 20718	. . . . . . . . 9 |- ( J e. (TopOn ` X ) -> J e. Top )
15	14	3ad2ant1 1082	. . . . . . . 8 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> J e. Top )
16		eqid 2622	. . . . . . . . 9 |- U. J = U. J
17	16	neii1 20910	. . . . . . . 8 |- ( ( J e. Top /\ n e. ( ( nei ` J ) ` { A } ) ) -> n C_ U. J )
18	15, 17	sylan 488	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ n e. ( ( nei ` J ) ` { A } ) ) -> n C_ U. J )
19		toponuni 20719	. . . . . . . . 9 |- ( J e. (TopOn ` X ) -> X = U. J )
20	19	3ad2ant1 1082	. . . . . . . 8 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> X = U. J )
21	20	adantr 481	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ n e. ( ( nei ` J ) ` { A } ) ) -> X = U. J )
22	18, 21	sseqtr4d 3642	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ n e. ( ( nei ` J ) ` { A } ) ) -> n C_ X )
23		elfm 21751	. . . . . . . 8 |- ( ( X e. J /\ L e. ( fBas ` Y ) /\ F : Y --> X ) -> ( n e. ( ( X FilMap F ) ` L ) <-> ( n C_ X /\ E. s e. L ( F " s ) C_ n ) ) )
24	5, 7, 8, 23	syl3anc 1326	. . . . . . 7 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( n e. ( ( X FilMap F ) ` L ) <-> ( n C_ X /\ E. s e. L ( F " s ) C_ n ) ) )
25	24	baibd 948	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ n C_ X ) -> ( n e. ( ( X FilMap F ) ` L ) <-> E. s e. L ( F " s ) C_ n ) )
26	22, 25	syldan 487	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ n e. ( ( nei ` J ) ` { A } ) ) -> ( n e. ( ( X FilMap F ) ` L ) <-> E. s e. L ( F " s ) C_ n ) )
27	26	ralbidva 2985	. . . 4 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( A. n e. ( ( nei ` J ) ` { A } ) n e. ( ( X FilMap F ) ` L ) <-> A. n e. ( ( nei ` J ) ` { A } ) E. s e. L ( F " s ) C_ n ) )
28	13, 27	syl5bb 272	. . 3 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( ( ( nei ` J ) ` { A } ) C_ ( ( X FilMap F ) ` L ) <-> A. n e. ( ( nei ` J ) ` { A } ) E. s e. L ( F " s ) C_ n ) )
29	28	anbi2d 740	. 2 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( ( A e. X /\ ( ( nei ` J ) ` { A } ) C_ ( ( X FilMap F ) ` L ) ) <-> ( A e. X /\ A. n e. ( ( nei ` J ) ` { A } ) E. s e. L ( F " s ) C_ n ) ) )
30	2, 12, 29	3bitrd 294	1 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( A e. ( ( J fLimf L ) ` F ) <-> ( A e. X /\ A. n e. ( ( nei ` J ) ` { A } ) E. s e. L ( F " s ) C_ n ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   {csn 4177   U.cuni 4436   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   fBascfbas 19734   Topctop 20698  TopOnctopon 20715   neicnei 20901   Filcfil 21649   FilMap cfm 21737   fLim cflim 21738   fLimf cflf 21739

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-map 7859  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-nei 20902  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744

This theorem is referenced by:  flfneii  21796  cnextcn  21871  cnextfres1  21872