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Theorem innei 19385
Description: The intersection of two neighborhoods of a set is also a neighborhood of the set. Proposition Vii of [BourbakiTop1] p. I.3 . (Contributed by FL, 28-Sep-2006.)
Assertion
Ref Expression
innei  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  M  e.  ( ( nei `  J
) `  S )
)  ->  ( N  i^i  M )  e.  ( ( nei `  J
) `  S )
)

Proof of Theorem innei
Dummy variables  g  h  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2460 . . . . 5  |-  U. J  =  U. J
21neii1 19366 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  N  C_  U. J )
3 ssinss1 3719 . . . 4  |-  ( N 
C_  U. J  ->  ( N  i^i  M )  C_  U. J )
42, 3syl 16 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  -> 
( N  i^i  M
)  C_  U. J )
543adant3 1011 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  M  e.  ( ( nei `  J
) `  S )
)  ->  ( N  i^i  M )  C_  U. J
)
6 neii2 19368 . . . . 5  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  E. h  e.  J  ( S  C_  h  /\  h  C_  N ) )
7 neii2 19368 . . . . 5  |-  ( ( J  e.  Top  /\  M  e.  ( ( nei `  J ) `  S ) )  ->  E. v  e.  J  ( S  C_  v  /\  v  C_  M ) )
86, 7anim12dan 834 . . . 4  |-  ( ( J  e.  Top  /\  ( N  e.  (
( nei `  J
) `  S )  /\  M  e.  (
( nei `  J
) `  S )
) )  ->  ( E. h  e.  J  ( S  C_  h  /\  h  C_  N )  /\  E. v  e.  J  ( S  C_  v  /\  v  C_  M ) ) )
9 inopn 19168 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  h  e.  J  /\  v  e.  J )  ->  ( h  i^i  v
)  e.  J )
1093expa 1191 . . . . . . . . . . 11  |-  ( ( ( J  e.  Top  /\  h  e.  J )  /\  v  e.  J
)  ->  ( h  i^i  v )  e.  J
)
11 ssin 3713 . . . . . . . . . . . . . 14  |-  ( ( S  C_  h  /\  S  C_  v )  <->  S  C_  (
h  i^i  v )
)
1211biimpi 194 . . . . . . . . . . . . 13  |-  ( ( S  C_  h  /\  S  C_  v )  ->  S  C_  ( h  i^i  v ) )
13 ss2in 3718 . . . . . . . . . . . . 13  |-  ( ( h  C_  N  /\  v  C_  M )  -> 
( h  i^i  v
)  C_  ( N  i^i  M ) )
1412, 13anim12i 566 . . . . . . . . . . . 12  |-  ( ( ( S  C_  h  /\  S  C_  v )  /\  ( h  C_  N  /\  v  C_  M
) )  ->  ( S  C_  ( h  i^i  v )  /\  (
h  i^i  v )  C_  ( N  i^i  M
) ) )
1514an4s 823 . . . . . . . . . . 11  |-  ( ( ( S  C_  h  /\  h  C_  N )  /\  ( S  C_  v  /\  v  C_  M
) )  ->  ( S  C_  ( h  i^i  v )  /\  (
h  i^i  v )  C_  ( N  i^i  M
) ) )
16 sseq2 3519 . . . . . . . . . . . . 13  |-  ( g  =  ( h  i^i  v )  ->  ( S  C_  g  <->  S  C_  (
h  i^i  v )
) )
17 sseq1 3518 . . . . . . . . . . . . 13  |-  ( g  =  ( h  i^i  v )  ->  (
g  C_  ( N  i^i  M )  <->  ( h  i^i  v )  C_  ( N  i^i  M ) ) )
1816, 17anbi12d 710 . . . . . . . . . . . 12  |-  ( g  =  ( h  i^i  v )  ->  (
( S  C_  g  /\  g  C_  ( N  i^i  M ) )  <-> 
( S  C_  (
h  i^i  v )  /\  ( h  i^i  v
)  C_  ( N  i^i  M ) ) ) )
1918rspcev 3207 . . . . . . . . . . 11  |-  ( ( ( h  i^i  v
)  e.  J  /\  ( S  C_  ( h  i^i  v )  /\  ( h  i^i  v
)  C_  ( N  i^i  M ) ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  ( N  i^i  M ) ) )
2010, 15, 19syl2an 477 . . . . . . . . . 10  |-  ( ( ( ( J  e. 
Top  /\  h  e.  J )  /\  v  e.  J )  /\  (
( S  C_  h  /\  h  C_  N )  /\  ( S  C_  v  /\  v  C_  M
) ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  ( N  i^i  M ) ) )
2120expr 615 . . . . . . . . 9  |-  ( ( ( ( J  e. 
Top  /\  h  e.  J )  /\  v  e.  J )  /\  ( S  C_  h  /\  h  C_  N ) )  -> 
( ( S  C_  v  /\  v  C_  M
)  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  ( N  i^i  M ) ) ) )
2221an32s 802 . . . . . . . 8  |-  ( ( ( ( J  e. 
Top  /\  h  e.  J )  /\  ( S  C_  h  /\  h  C_  N ) )  /\  v  e.  J )  ->  ( ( S  C_  v  /\  v  C_  M
)  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  ( N  i^i  M ) ) ) )
2322rexlimdva 2948 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  h  e.  J )  /\  ( S  C_  h  /\  h  C_  N
) )  ->  ( E. v  e.  J  ( S  C_  v  /\  v  C_  M )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  ( N  i^i  M ) ) ) )
2423ex 434 . . . . . 6  |-  ( ( J  e.  Top  /\  h  e.  J )  ->  ( ( S  C_  h  /\  h  C_  N
)  ->  ( E. v  e.  J  ( S  C_  v  /\  v  C_  M )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  ( N  i^i  M ) ) ) ) )
2524rexlimdva 2948 . . . . 5  |-  ( J  e.  Top  ->  ( E. h  e.  J  ( S  C_  h  /\  h  C_  N )  -> 
( E. v  e.  J  ( S  C_  v  /\  v  C_  M
)  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  ( N  i^i  M ) ) ) ) )
2625imp32 433 . . . 4  |-  ( ( J  e.  Top  /\  ( E. h  e.  J  ( S  C_  h  /\  h  C_  N )  /\  E. v  e.  J  ( S  C_  v  /\  v  C_  M ) ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  ( N  i^i  M ) ) )
278, 26syldan 470 . . 3  |-  ( ( J  e.  Top  /\  ( N  e.  (
( nei `  J
) `  S )  /\  M  e.  (
( nei `  J
) `  S )
) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  ( N  i^i  M ) ) )
28273impb 1187 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  M  e.  ( ( nei `  J
) `  S )
)  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  ( N  i^i  M ) ) )
291neiss2 19361 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  U. J )
301isnei 19363 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  U. J )  ->  ( ( N  i^i  M )  e.  ( ( nei `  J
) `  S )  <->  ( ( N  i^i  M
)  C_  U. J  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  ( N  i^i  M ) ) ) ) )
3129, 30syldan 470 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  -> 
( ( N  i^i  M )  e.  ( ( nei `  J ) `
 S )  <->  ( ( N  i^i  M )  C_  U. J  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  ( N  i^i  M ) ) ) ) )
32313adant3 1011 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  M  e.  ( ( nei `  J
) `  S )
)  ->  ( ( N  i^i  M )  e.  ( ( nei `  J
) `  S )  <->  ( ( N  i^i  M
)  C_  U. J  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  ( N  i^i  M ) ) ) ) )
335, 28, 32mpbir2and 915 1  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  M  e.  ( ( nei `  J
) `  S )
)  ->  ( N  i^i  M )  e.  ( ( nei `  J
) `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   E.wrex 2808    i^i cin 3468    C_ wss 3469   U.cuni 4238   ` cfv 5579   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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  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-top 19159  df-nei 19358
This theorem is referenced by:  neifil  20109  neificl  29836
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