Theorem trnei 21696

Description: The trace, over a set A, of the filter of the neighborhoods of a point P is a filter iff P belongs to the closure of A. (This is trfil2 21691 applied to a filter of neighborhoods.) (Contributed by FL, 15-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
trnei	|- ( ( J e. (TopOn ` Y ) /\ A C_ Y /\ P e. Y ) -> ( P e. ( ( cls ` J ) ` A ) <-> ( ( ( nei ` J ) ` { P } ) ↾t A ) e. ( Fil ` A ) ) )

Proof of Theorem trnei

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		topontop 20718	. . . 4 |- ( J e. (TopOn ` Y ) -> J e. Top )
2	1	3ad2ant1 1082	. . 3 |- ( ( J e. (TopOn ` Y ) /\ A C_ Y /\ P e. Y ) -> J e. Top )
3		simp2 1062	. . . 4 |- ( ( J e. (TopOn ` Y ) /\ A C_ Y /\ P e. Y ) -> A C_ Y )
4		toponuni 20719	. . . . 5 |- ( J e. (TopOn ` Y ) -> Y = U. J )
5	4	3ad2ant1 1082	. . . 4 |- ( ( J e. (TopOn ` Y ) /\ A C_ Y /\ P e. Y ) -> Y = U. J )
6	3, 5	sseqtrd 3641	. . 3 |- ( ( J e. (TopOn ` Y ) /\ A C_ Y /\ P e. Y ) -> A C_ U. J )
7		simp3 1063	. . . 4 |- ( ( J e. (TopOn ` Y ) /\ A C_ Y /\ P e. Y ) -> P e. Y )
8	7, 5	eleqtrd 2703	. . 3 |- ( ( J e. (TopOn ` Y ) /\ A C_ Y /\ P e. Y ) -> P e. U. J )
9		eqid 2622	. . . 4 |- U. J = U. J
10	9	neindisj2 20927	. . 3 |- ( ( J e. Top /\ A C_ U. J /\ P e. U. J ) -> ( P e. ( ( cls ` J ) ` A ) <-> A. v e. ( ( nei ` J ) ` { P } ) ( v i^i A ) =/= (/) ) )
11	2, 6, 8, 10	syl3anc 1326	. 2 |- ( ( J e. (TopOn ` Y ) /\ A C_ Y /\ P e. Y ) -> ( P e. ( ( cls ` J ) ` A ) <-> A. v e. ( ( nei ` J ) ` { P } ) ( v i^i A ) =/= (/) ) )
12		simp1 1061	. . . 4 |- ( ( J e. (TopOn ` Y ) /\ A C_ Y /\ P e. Y ) -> J e. (TopOn ` Y ) )
13	7	snssd 4340	. . . 4 |- ( ( J e. (TopOn ` Y ) /\ A C_ Y /\ P e. Y ) -> { P } C_ Y )
14		snnzg 4308	. . . . 5 |- ( P e. Y -> { P } =/= (/) )
15	14	3ad2ant3 1084	. . . 4 |- ( ( J e. (TopOn ` Y ) /\ A C_ Y /\ P e. Y ) -> { P } =/= (/) )
16		neifil 21684	. . . 4 |- ( ( J e. (TopOn ` Y ) /\ { P } C_ Y /\ { P } =/= (/) ) -> ( ( nei ` J ) ` { P } ) e. ( Fil ` Y ) )
17	12, 13, 15, 16	syl3anc 1326	. . 3 |- ( ( J e. (TopOn ` Y ) /\ A C_ Y /\ P e. Y ) -> ( ( nei ` J ) ` { P } ) e. ( Fil ` Y ) )
18		trfil2 21691	. . 3 |- ( ( ( ( nei ` J ) ` { P } ) e. ( Fil ` Y ) /\ A C_ Y ) -> ( ( ( ( nei ` J ) ` { P } ) ↾t A ) e. ( Fil ` A ) <-> A. v e. ( ( nei ` J ) ` { P } ) ( v i^i A ) =/= (/) ) )
19	17, 3, 18	syl2anc 693	. 2 |- ( ( J e. (TopOn ` Y ) /\ A C_ Y /\ P e. Y ) -> ( ( ( ( nei ` J ) ` { P } ) ↾t A ) e. ( Fil ` A ) <-> A. v e. ( ( nei ` J ) ` { P } ) ( v i^i A ) =/= (/) ) )
20	11, 19	bitr4d 271	1 |- ( ( J e. (TopOn ` Y ) /\ A C_ Y /\ P e. Y ) -> ( P e. ( ( cls ` J ) ` A ) <-> ( ( ( nei ` J ) ` { P } ) ↾t A ) e. ( Fil ` A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   U.cuni 4436   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698  TopOnctopon 20715   clsccl 20822   neicnei 20901   Filcfil 21649

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-int 4476  df-iun 4522  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-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-rest 16083  df-fbas 19743  df-top 20699  df-topon 20716  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-fil 21650

This theorem is referenced by:  flfcntr  21847  cnextfun  21868  cnextfvval  21869  cnextf  21870  cnextcn  21871  cnextfres1  21872  cnextucn  22107  ucnextcn  22108  limcflflem  23644  rrhre  30065