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Theorem neificl 26467
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 732 . . 3  |-  ( ( N  C_  ( ( nei `  J ) `  S )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) )  ->  N  e.  Fin )
2 innei 16862 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S )  /\  y  e.  ( ( nei `  J
) `  S )
)  ->  ( x  i^i  y )  e.  ( ( nei `  J
) `  S )
)
323expib 1154 . . . . . . 7  |-  ( J  e.  Top  ->  (
( x  e.  ( ( nei `  J
) `  S )  /\  y  e.  (
( nei `  J
) `  S )
)  ->  ( x  i^i  y )  e.  ( ( nei `  J
) `  S )
) )
43ralrimivv 2634 . . . . . 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 7133 . . . . . 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 188 . . . . 5  |-  ( J  e.  Top  ->  A. x
( ( x  C_  ( ( nei `  J
) `  S )  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  ( ( nei `  J
) `  S )
) )
7 sseq1 3199 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  C_  ( ( nei `  J ) `  S )  <->  N  C_  (
( nei `  J
) `  S )
) )
8 neeq1 2454 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  =/=  (/)  <->  N  =/=  (/) ) )
9 eleq1 2343 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  e.  Fin  <->  N  e.  Fin ) )
107, 8, 93anbi123d 1252 . . . . . . . 8  |-  ( x  =  N  ->  (
( x  C_  (
( nei `  J
) `  S )  /\  x  =/=  (/)  /\  x  e.  Fin )  <->  ( N  C_  ( ( nei `  J
) `  S )  /\  N  =/=  (/)  /\  N  e.  Fin ) ) )
11 3ancomb 943 . . . . . . . . 9  |-  ( ( N  C_  ( ( nei `  J ) `  S )  /\  N  =/=  (/)  /\  N  e. 
Fin )  <->  ( N  C_  ( ( nei `  J
) `  S )  /\  N  e.  Fin  /\  N  =/=  (/) ) )
12 3anass 938 . . . . . . . . 9  |-  ( ( N  C_  ( ( nei `  J ) `  S )  /\  N  e.  Fin  /\  N  =/=  (/) )  <->  ( N  C_  ( ( nei `  J
) `  S )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) ) )
1311, 12bitri 240 . . . . . . . 8  |-  ( ( N  C_  ( ( nei `  J ) `  S )  /\  N  =/=  (/)  /\  N  e. 
Fin )  <->  ( N  C_  ( ( nei `  J
) `  S )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) ) )
1410, 13syl6bb 252 . . . . . . 7  |-  ( x  =  N  ->  (
( x  C_  (
( nei `  J
) `  S )  /\  x  =/=  (/)  /\  x  e.  Fin )  <->  ( N  C_  ( ( nei `  J
) `  S )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) ) ) )
15 inteq 3865 . . . . . . . 8  |-  ( x  =  N  ->  |^| x  =  |^| N )
1615eleq1d 2349 . . . . . . 7  |-  ( x  =  N  ->  ( |^| x  e.  (
( nei `  J
) `  S )  <->  |^| N  e.  ( ( nei `  J ) `
 S ) ) )
1714, 16imbi12d 311 . . . . . 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 2868 . . . . 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 28 . . . 4  |-  ( N  e.  Fin  ->  ( J  e.  Top  ->  (
( N  C_  (
( nei `  J
) `  S )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) )  ->  |^| N  e.  ( ( nei `  J
) `  S )
) ) )
2019com3l 75 . . 3  |-  ( J  e.  Top  ->  (
( N  C_  (
( nei `  J
) `  S )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) )  ->  ( N  e. 
Fin  ->  |^| N  e.  ( ( nei `  J
) `  S )
) ) )
211, 20mpdi 38 . 2  |-  ( J  e.  Top  ->  (
( N  C_  (
( nei `  J
) `  S )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) )  ->  |^| N  e.  ( ( nei `  J
) `  S )
) )
2221impl 603 1  |-  ( ( ( J  e.  Top  /\  N  C_  ( ( nei `  J ) `  S ) )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) )  ->  |^| N  e.  ( ( nei `  J
) `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   A.wal 1527    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    i^i cin 3151    C_ wss 3152   (/)c0 3455   |^|cint 3862   ` cfv 5255   Fincfn 6863   Topctop 16631   neicnei 16834
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-fin 6867  df-top 16636  df-nei 16835
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