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Theorem innei 16878
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 2296 . . . . 5  |-  U. J  =  U. J
21neii1 16859 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  N  C_  U. J )
3 ssinss1 3410 . . . 4  |-  ( N 
C_  U. J  ->  ( N  i^i  M )  C_  U. J )
42, 3syl 15 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  -> 
( N  i^i  M
)  C_  U. J )
543adant3 975 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  M  e.  ( ( nei `  J
) `  S )
)  ->  ( N  i^i  M )  C_  U. J
)
6 neii2 16861 . . . . 5  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  E. h  e.  J  ( S  C_  h  /\  h  C_  N ) )
7 neii2 16861 . . . . 5  |-  ( ( J  e.  Top  /\  M  e.  ( ( nei `  J ) `  S ) )  ->  E. v  e.  J  ( S  C_  v  /\  v  C_  M ) )
86, 7anim12dan 810 . . . 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 16661 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  h  e.  J  /\  v  e.  J )  ->  ( h  i^i  v
)  e.  J )
1093expa 1151 . . . . . . . . . . 11  |-  ( ( ( J  e.  Top  /\  h  e.  J )  /\  v  e.  J
)  ->  ( h  i^i  v )  e.  J
)
11 ssin 3404 . . . . . . . . . . . . . 14  |-  ( ( S  C_  h  /\  S  C_  v )  <->  S  C_  (
h  i^i  v )
)
1211biimpi 186 . . . . . . . . . . . . 13  |-  ( ( S  C_  h  /\  S  C_  v )  ->  S  C_  ( h  i^i  v ) )
13 ss2in 3409 . . . . . . . . . . . . 13  |-  ( ( h  C_  N  /\  v  C_  M )  -> 
( h  i^i  v
)  C_  ( N  i^i  M ) )
1412, 13anim12i 549 . . . . . . . . . . . 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 799 . . . . . . . . . . 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 3213 . . . . . . . . . . . . 13  |-  ( g  =  ( h  i^i  v )  ->  ( S  C_  g  <->  S  C_  (
h  i^i  v )
) )
17 sseq1 3212 . . . . . . . . . . . . 13  |-  ( g  =  ( h  i^i  v )  ->  (
g  C_  ( N  i^i  M )  <->  ( h  i^i  v )  C_  ( N  i^i  M ) ) )
1816, 17anbi12d 691 . . . . . . . . . . . 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 2897 . . . . . . . . . . 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 463 . . . . . . . . . 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 598 . . . . . . . . 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 779 . . . . . . . 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 2680 . . . . . . 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 423 . . . . . 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 2680 . . . . 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 422 . . . 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 456 . . 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 1147 . 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 16854 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  U. J )
301isnei 16856 . . . 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 456 . . 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 975 . 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 888 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557    i^i cin 3164    C_ wss 3165   U.cuni 3843   ` cfv 5271   Topctop 16647   neicnei 16850
This theorem is referenced by:  neifil  17591  islimrs4  25685  neificl  26570
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 16652  df-nei 16851
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