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Theorem neificl 28790
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 18854 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S )  /\  y  e.  ( ( nei `  J
) `  S )
)  ->  ( x  i^i  y )  e.  ( ( nei `  J
) `  S )
)
323expib 1191 . . . . . . 7  |-  ( J  e.  Top  ->  (
( x  e.  ( ( nei `  J
) `  S )  /\  y  e.  (
( nei `  J
) `  S )
)  ->  ( x  i^i  y )  e.  ( ( nei `  J
) `  S )
) )
43ralrimivv 2906 . . . . . 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 7692 . . . . . 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 3478 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  C_  ( ( nei `  J ) `  S )  <->  N  C_  (
( nei `  J
) `  S )
) )
8 neeq1 2729 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  =/=  (/)  <->  N  =/=  (/) ) )
9 eleq1 2523 . . . . . . . . 9  |-  ( x  =  N  ->  (
x  e.  Fin  <->  N  e.  Fin ) )
107, 8, 93anbi123d 1290 . . . . . . . 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 4232 . . . . . . . 8  |-  ( x  =  N  ->  |^| x  =  |^| N )
1615eleq1d 2520 . . . . . . 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 3156 . . . . 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 1368    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795    i^i cin 3428    C_ wss 3429   (/)c0 3738   |^|cint 4229   ` cfv 5519   Fincfn 7413   Topctop 18623   neicnei 18826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-en 7414  df-fin 7417  df-top 18628  df-nei 18827
This theorem is referenced by: (None)
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