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Theorem nllyi 20567
Description: The property of an n-locally  A topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nllyi  |-  ( ( J  e. 𝑛Locally  A  /\  U  e.  J  /\  P  e.  U )  ->  E. u  e.  ( ( nei `  J
) `  { P } ) ( u 
C_  U  /\  ( Jt  u )  e.  A
) )
Distinct variable groups:    u, A    u, P    u, U    u, J

Proof of Theorem nllyi
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnlly 20561 . . . 4  |-  ( J  e. 𝑛Locally  A  <->  ( J  e. 
Top  /\  A. x  e.  J  A. y  e.  x  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P x ) ( Jt  u )  e.  A
) )
21simprbi 471 . . 3  |-  ( J  e. 𝑛Locally  A  ->  A. x  e.  J  A. y  e.  x  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P x ) ( Jt  u )  e.  A
)
3 pweq 3945 . . . . . . 7  |-  ( x  =  U  ->  ~P x  =  ~P U
)
43ineq2d 3625 . . . . . 6  |-  ( x  =  U  ->  (
( ( nei `  J
) `  { y } )  i^i  ~P x )  =  ( ( ( nei `  J
) `  { y } )  i^i  ~P U ) )
54rexeqdv 2980 . . . . 5  |-  ( x  =  U  ->  ( E. u  e.  (
( ( nei `  J
) `  { y } )  i^i  ~P x ) ( Jt  u )  e.  A  <->  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P U ) ( Jt  u )  e.  A
) )
65raleqbi1dv 2981 . . . 4  |-  ( x  =  U  ->  ( A. y  e.  x  E. u  e.  (
( ( nei `  J
) `  { y } )  i^i  ~P x ) ( Jt  u )  e.  A  <->  A. y  e.  U  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P U ) ( Jt  u )  e.  A
) )
76rspccva 3135 . . 3  |-  ( ( A. x  e.  J  A. y  e.  x  E. u  e.  (
( ( nei `  J
) `  { y } )  i^i  ~P x ) ( Jt  u )  e.  A  /\  U  e.  J )  ->  A. y  e.  U  E. u  e.  (
( ( nei `  J
) `  { y } )  i^i  ~P U ) ( Jt  u )  e.  A )
82, 7sylan 479 . 2  |-  ( ( J  e. 𝑛Locally  A  /\  U  e.  J )  ->  A. y  e.  U  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P U ) ( Jt  u )  e.  A
)
9 elin 3608 . . . . . . 7  |-  ( u  e.  ( ( ( nei `  J ) `
 { y } )  i^i  ~P U
)  <->  ( u  e.  ( ( nei `  J
) `  { y } )  /\  u  e.  ~P U ) )
10 sneq 3969 . . . . . . . . . 10  |-  ( y  =  P  ->  { y }  =  { P } )
1110fveq2d 5883 . . . . . . . . 9  |-  ( y  =  P  ->  (
( nei `  J
) `  { y } )  =  ( ( nei `  J
) `  { P } ) )
1211eleq2d 2534 . . . . . . . 8  |-  ( y  =  P  ->  (
u  e.  ( ( nei `  J ) `
 { y } )  <->  u  e.  (
( nei `  J
) `  { P } ) ) )
13 selpw 3949 . . . . . . . . 9  |-  ( u  e.  ~P U  <->  u  C_  U
)
1413a1i 11 . . . . . . . 8  |-  ( y  =  P  ->  (
u  e.  ~P U  <->  u 
C_  U ) )
1512, 14anbi12d 725 . . . . . . 7  |-  ( y  =  P  ->  (
( u  e.  ( ( nei `  J
) `  { y } )  /\  u  e.  ~P U )  <->  ( u  e.  ( ( nei `  J
) `  { P } )  /\  u  C_  U ) ) )
169, 15syl5bb 265 . . . . . 6  |-  ( y  =  P  ->  (
u  e.  ( ( ( nei `  J
) `  { y } )  i^i  ~P U )  <->  ( u  e.  ( ( nei `  J
) `  { P } )  /\  u  C_  U ) ) )
1716anbi1d 719 . . . . 5  |-  ( y  =  P  ->  (
( u  e.  ( ( ( nei `  J
) `  { y } )  i^i  ~P U )  /\  ( Jt  u )  e.  A
)  <->  ( ( u  e.  ( ( nei `  J ) `  { P } )  /\  u  C_  U )  /\  ( Jt  u )  e.  A
) ) )
18 anass 661 . . . . 5  |-  ( ( ( u  e.  ( ( nei `  J
) `  { P } )  /\  u  C_  U )  /\  ( Jt  u )  e.  A
)  <->  ( u  e.  ( ( nei `  J
) `  { P } )  /\  (
u  C_  U  /\  ( Jt  u )  e.  A
) ) )
1917, 18syl6bb 269 . . . 4  |-  ( y  =  P  ->  (
( u  e.  ( ( ( nei `  J
) `  { y } )  i^i  ~P U )  /\  ( Jt  u )  e.  A
)  <->  ( u  e.  ( ( nei `  J
) `  { P } )  /\  (
u  C_  U  /\  ( Jt  u )  e.  A
) ) ) )
2019rexbidv2 2888 . . 3  |-  ( y  =  P  ->  ( E. u  e.  (
( ( nei `  J
) `  { y } )  i^i  ~P U ) ( Jt  u )  e.  A  <->  E. u  e.  ( ( nei `  J
) `  { P } ) ( u 
C_  U  /\  ( Jt  u )  e.  A
) ) )
2120rspccva 3135 . 2  |-  ( ( A. y  e.  U  E. u  e.  (
( ( nei `  J
) `  { y } )  i^i  ~P U ) ( Jt  u )  e.  A  /\  P  e.  U )  ->  E. u  e.  ( ( nei `  J
) `  { P } ) ( u 
C_  U  /\  ( Jt  u )  e.  A
) )
228, 21stoic3 1668 1  |-  ( ( J  e. 𝑛Locally  A  /\  U  e.  J  /\  P  e.  U )  ->  E. u  e.  ( ( nei `  J
) `  { P } ) ( u 
C_  U  /\  ( Jt  u )  e.  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   {csn 3959   ` cfv 5589  (class class class)co 6308   ↾t crest 15397   Topctop 19994   neicnei 20190  𝑛Locally cnlly 20557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-ov 6311  df-nlly 20559
This theorem is referenced by:  nlly2i  20568  llycmpkgen  20644
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