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Theorem neificl 31528
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 756 . . 3  |-  ( ( N  C_  ( ( nei `  J ) `  S )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) )  ->  N  e.  Fin )
2 innei 19919 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S )  /\  y  e.  ( ( nei `  J
) `  S )
)  ->  ( x  i^i  y )  e.  ( ( nei `  J
) `  S )
)
323expib 1200 . . . . . . 7  |-  ( J  e.  Top  ->  (
( x  e.  ( ( nei `  J
) `  S )  /\  y  e.  (
( nei `  J
) `  S )
)  ->  ( x  i^i  y )  e.  ( ( nei `  J
) `  S )
) )
43ralrimivv 2824 . . . . . 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 7831 . . . . . 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 3463 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  C_  ( ( nei `  J ) `  S )  <->  N  C_  (
( nei `  J
) `  S )
) )
8 neeq1 2684 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  =/=  (/)  <->  N  =/=  (/) ) )
9 eleq1 2474 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  e.  Fin  <->  N  e.  Fin ) )
107, 8, 93anbi123d 1301 . . . . . . . 8  |-  ( x  =  N  ->  (
( x  C_  (
( nei `  J
) `  S )  /\  x  =/=  (/)  /\  x  e.  Fin )  <->  ( N  C_  ( ( nei `  J
) `  S )  /\  N  =/=  (/)  /\  N  e.  Fin ) ) )
11 3ancomb 983 . . . . . . . . 9  |-  ( ( N  C_  ( ( nei `  J ) `  S )  /\  N  =/=  (/)  /\  N  e. 
Fin )  <->  ( N  C_  ( ( nei `  J
) `  S )  /\  N  e.  Fin  /\  N  =/=  (/) ) )
12 3anass 978 . . . . . . . . 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 4230 . . . . . . . 8  |-  ( x  =  N  ->  |^| x  =  |^| N )
1615eleq1d 2471 . . . . . . 7  |-  ( x  =  N  ->  ( |^| x  e.  (
( nei `  J
) `  S )  <->  |^| N  e.  ( ( nei `  J ) `
 S ) ) )
1714, 16imbi12d 318 . . . . . 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 3144 . . . . 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 30 . . . 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 40 . 2  |-  ( J  e.  Top  ->  (
( N  C_  (
( nei `  J
) `  S )  /\  ( N  e.  Fin  /\  N  =/=  (/) ) )  ->  |^| N  e.  ( ( nei `  J
) `  S )
) )
2221impl 618 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 367    /\ w3a 974   A.wal 1403    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754    i^i cin 3413    C_ wss 3414   (/)c0 3738   |^|cint 4227   ` cfv 5569   Fincfn 7554   Topctop 19686   neicnei 19891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-fin 7558  df-top 19691  df-nei 19892
This theorem is referenced by: (None)
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