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Theorem innei 12957
Description: The intersection of two neighborhoods of a set is also a neighborhood of the set. Generalization to subsets of Property Vii of [BourbakiTop1] p. I.3 for binary intersections. (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 2170 . . . . 5  |-  U. J  =  U. J
21neii1 12941 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  N  C_  U. J )
3 ssinss1 3356 . . . 4  |-  ( N 
C_  U. J  ->  ( N  i^i  M )  C_  U. J )
42, 3syl 14 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  -> 
( N  i^i  M
)  C_  U. J )
543adant3 1012 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  M  e.  ( ( nei `  J
) `  S )
)  ->  ( N  i^i  M )  C_  U. J
)
6 neii2 12943 . . . . 5  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  E. h  e.  J  ( S  C_  h  /\  h  C_  N ) )
7 neii2 12943 . . . . 5  |-  ( ( J  e.  Top  /\  M  e.  ( ( nei `  J ) `  S ) )  ->  E. v  e.  J  ( S  C_  v  /\  v  C_  M ) )
86, 7anim12dan 595 . . . 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 12795 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  h  e.  J  /\  v  e.  J )  ->  ( h  i^i  v
)  e.  J )
1093expa 1198 . . . . . . . . . 10  |-  ( ( ( J  e.  Top  /\  h  e.  J )  /\  v  e.  J
)  ->  ( h  i^i  v )  e.  J
)
11 ssin 3349 . . . . . . . . . . . . 13  |-  ( ( S  C_  h  /\  S  C_  v )  <->  S  C_  (
h  i^i  v )
)
1211biimpi 119 . . . . . . . . . . . 12  |-  ( ( S  C_  h  /\  S  C_  v )  ->  S  C_  ( h  i^i  v ) )
13 ss2in 3355 . . . . . . . . . . . 12  |-  ( ( h  C_  N  /\  v  C_  M )  -> 
( h  i^i  v
)  C_  ( N  i^i  M ) )
1412, 13anim12i 336 . . . . . . . . . . 11  |-  ( ( ( 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 583 . . . . . . . . . 10  |-  ( ( ( 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 3171 . . . . . . . . . . . 12  |-  ( g  =  ( h  i^i  v )  ->  ( S  C_  g  <->  S  C_  (
h  i^i  v )
) )
17 sseq1 3170 . . . . . . . . . . . 12  |-  ( g  =  ( h  i^i  v )  ->  (
g  C_  ( N  i^i  M )  <->  ( h  i^i  v )  C_  ( N  i^i  M ) ) )
1816, 17anbi12d 470 . . . . . . . . . . 11  |-  ( 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 2834 . . . . . . . . . 10  |-  ( ( ( 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 287 . . . . . . . . 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 ) ) )
2120expr 373 . . . . . . . 8  |-  ( ( ( ( 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 563 . . . . . . 7  |-  ( ( ( ( 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 2587 . . . . . 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 ) ) ) )
2423rexlimdva2 2590 . . . . 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 ) ) ) ) )
2524imp32 255 . . . 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 ) ) )
268, 25syldan 280 . . 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 ) ) )
27263impb 1194 . 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 ) ) )
281neiss2 12936 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  U. J )
291isnei 12938 . . . 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 ) ) ) ) )
3028, 29syldan 280 . . 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 ) ) ) ) )
31303adant3 1012 . 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 ) ) ) ) )
325, 27, 31mpbir2and 939 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    /\ wa 103    <-> wb 104    /\ w3a 973    = wceq 1348    e. wcel 2141   E.wrex 2449    i^i cin 3120    C_ wss 3121   U.cuni 3796   ` cfv 5198   Topctop 12789   neicnei 12932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-top 12790  df-nei 12933
This theorem is referenced by: (None)
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