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Theorem neificl 28602
Description: Neighborhoods are closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Nov-2013.)
Assertion
Ref Expression
neificl  |-  ( ( ( J  e.  Top  /\  N  C_  ( ( nei `  J ) `  S ) )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) )  ->  |^| N  e.  ( ( nei `  J
) `  S )
)

Proof of Theorem neificl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 755 . . 3  |-  ( ( N  C_  ( ( nei `  J ) `  S )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) )  ->  N  e.  Fin )
2 innei 18704 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S )  /\  y  e.  ( ( nei `  J
) `  S )
)  ->  ( x  i^i  y )  e.  ( ( nei `  J
) `  S )
)
323expib 1190 . . . . . . 7  |-  ( J  e.  Top  ->  (
( x  e.  ( ( nei `  J
) `  S )  /\  y  e.  (
( nei `  J
) `  S )
)  ->  ( x  i^i  y )  e.  ( ( nei `  J
) `  S )
) )
43ralrimivv 2802 . . . . . 6  |-  ( J  e.  Top  ->  A. x  e.  ( ( nei `  J
) `  S ) A. y  e.  (
( nei `  J
) `  S )
( x  i^i  y
)  e.  ( ( nei `  J ) `
 S ) )
5 fiint 7580 . . . . . 6  |-  ( A. x  e.  ( ( nei `  J ) `  S ) A. y  e.  ( ( nei `  J
) `  S )
( x  i^i  y
)  e.  ( ( nei `  J ) `
 S )  <->  A. x
( ( x  C_  ( ( nei `  J
) `  S )  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  ( ( nei `  J
) `  S )
) )
64, 5sylib 196 . . . . 5  |-  ( J  e.  Top  ->  A. x
( ( x  C_  ( ( nei `  J
) `  S )  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  ( ( nei `  J
) `  S )
) )
7 sseq1 3372 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  C_  ( ( nei `  J ) `  S )  <->  N  C_  (
( nei `  J
) `  S )
) )
8 neeq1 2611 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  =/=  (/)  <->  N  =/=  (/) ) )
9 eleq1 2498 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  e.  Fin  <->  N  e.  Fin ) )
107, 8, 93anbi123d 1289 . . . . . . . 8  |-  ( x  =  N  ->  (
( x  C_  (
( nei `  J
) `  S )  /\  x  =/=  (/)  /\  x  e.  Fin )  <->  ( N  C_  ( ( nei `  J
) `  S )  /\  N  =/=  (/)  /\  N  e.  Fin ) ) )
11 3ancomb 974 . . . . . . . . 9  |-  ( ( N  C_  ( ( nei `  J ) `  S )  /\  N  =/=  (/)  /\  N  e. 
Fin )  <->  ( N  C_  ( ( nei `  J
) `  S )  /\  N  e.  Fin  /\  N  =/=  (/) ) )
12 3anass 969 . . . . . . . . 9  |-  ( ( N  C_  ( ( nei `  J ) `  S )  /\  N  e.  Fin  /\  N  =/=  (/) )  <->  ( N  C_  ( ( nei `  J
) `  S )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) ) )
1311, 12bitri 249 . . . . . . . 8  |-  ( ( N  C_  ( ( nei `  J ) `  S )  /\  N  =/=  (/)  /\  N  e. 
Fin )  <->  ( N  C_  ( ( nei `  J
) `  S )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) ) )
1410, 13syl6bb 261 . . . . . . 7  |-  ( x  =  N  ->  (
( x  C_  (
( nei `  J
) `  S )  /\  x  =/=  (/)  /\  x  e.  Fin )  <->  ( N  C_  ( ( nei `  J
) `  S )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) ) ) )
15 inteq 4126 . . . . . . . 8  |-  ( x  =  N  ->  |^| x  =  |^| N )
1615eleq1d 2504 . . . . . . 7  |-  ( x  =  N  ->  ( |^| x  e.  (
( nei `  J
) `  S )  <->  |^| N  e.  ( ( nei `  J ) `
 S ) ) )
1714, 16imbi12d 320 . . . . . 6  |-  ( x  =  N  ->  (
( ( x  C_  ( ( nei `  J
) `  S )  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  ( ( nei `  J
) `  S )
)  <->  ( ( N 
C_  ( ( nei `  J ) `  S
)  /\  ( N  e.  Fin  /\  N  =/=  (/) ) )  ->  |^| N  e.  ( ( nei `  J
) `  S )
) ) )
1817spcgv 3052 . . . . 5  |-  ( N  e.  Fin  ->  ( A. x ( ( x 
C_  ( ( nei `  J ) `  S
)  /\  x  =/=  (/) 
/\  x  e.  Fin )  ->  |^| x  e.  ( ( nei `  J
) `  S )
)  ->  ( ( N  C_  ( ( nei `  J ) `  S
)  /\  ( N  e.  Fin  /\  N  =/=  (/) ) )  ->  |^| N  e.  ( ( nei `  J
) `  S )
) ) )
196, 18syl5 32 . . . 4  |-  ( N  e.  Fin  ->  ( J  e.  Top  ->  (
( N  C_  (
( nei `  J
) `  S )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) )  ->  |^| N  e.  ( ( nei `  J
) `  S )
) ) )
2019com3l 81 . . 3  |-  ( J  e.  Top  ->  (
( N  C_  (
( nei `  J
) `  S )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) )  ->  ( N  e. 
Fin  ->  |^| N  e.  ( ( nei `  J
) `  S )
) ) )
211, 20mpdi 42 . 2  |-  ( J  e.  Top  ->  (
( N  C_  (
( nei `  J
) `  S )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) )  ->  |^| N  e.  ( ( nei `  J
) `  S )
) )
2221impl 620 1  |-  ( ( ( J  e.  Top  /\  N  C_  ( ( nei `  J ) `  S ) )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) )  ->  |^| N  e.  ( ( nei `  J
) `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710    i^i cin 3322    C_ wss 3323   (/)c0 3632   |^|cint 4123   ` cfv 5413   Fincfn 7302   Topctop 18473   neicnei 18676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-fin 7306  df-top 18478  df-nei 18677
This theorem is referenced by: (None)
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