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Theorem neificl 29700
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 19385 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S )  /\  y  e.  ( ( nei `  J
) `  S )
)  ->  ( x  i^i  y )  e.  ( ( nei `  J
) `  S )
)
323expib 1194 . . . . . . 7  |-  ( J  e.  Top  ->  (
( x  e.  ( ( nei `  J
) `  S )  /\  y  e.  (
( nei `  J
) `  S )
)  ->  ( x  i^i  y )  e.  ( ( nei `  J
) `  S )
) )
43ralrimivv 2877 . . . . . 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 7786 . . . . . 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 3518 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  C_  ( ( nei `  J ) `  S )  <->  N  C_  (
( nei `  J
) `  S )
) )
8 neeq1 2741 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  =/=  (/)  <->  N  =/=  (/) ) )
9 eleq1 2532 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  e.  Fin  <->  N  e.  Fin ) )
107, 8, 93anbi123d 1294 . . . . . . . 8  |-  ( x  =  N  ->  (
( x  C_  (
( nei `  J
) `  S )  /\  x  =/=  (/)  /\  x  e.  Fin )  <->  ( N  C_  ( ( nei `  J
) `  S )  /\  N  =/=  (/)  /\  N  e.  Fin ) ) )
11 3ancomb 977 . . . . . . . . 9  |-  ( ( N  C_  ( ( nei `  J ) `  S )  /\  N  =/=  (/)  /\  N  e. 
Fin )  <->  ( N  C_  ( ( nei `  J
) `  S )  /\  N  e.  Fin  /\  N  =/=  (/) ) )
12 3anass 972 . . . . . . . . 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 4278 . . . . . . . 8  |-  ( x  =  N  ->  |^| x  =  |^| N )
1615eleq1d 2529 . . . . . . 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 3191 . . . . 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 968   A.wal 1372    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807    i^i cin 3468    C_ wss 3469   (/)c0 3778   |^|cint 4275   ` cfv 5579   Fincfn 7506   Topctop 19154   neicnei 19357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-fin 7510  df-top 19159  df-nei 19358
This theorem is referenced by: (None)
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