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Theorem nllyi 19782
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 19776 . . . . 5  |-  ( 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 464 . . . 4  |-  ( 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 4013 . . . . . . . 8  |-  ( x  =  U  ->  ~P x  =  ~P U
)
43ineq2d 3700 . . . . . . 7  |-  ( x  =  U  ->  (
( ( nei `  J
) `  { y } )  i^i  ~P x )  =  ( ( ( nei `  J
) `  { y } )  i^i  ~P U ) )
54rexeqdv 3065 . . . . . 6  |-  ( 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 3066 . . . . 5  |-  ( 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 3213 . . . 4  |-  ( ( 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 471 . . 3  |-  ( ( 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 3687 . . . . . . . 8  |-  ( u  e.  ( ( ( nei `  J ) `
 { y } )  i^i  ~P U
)  <->  ( u  e.  ( ( nei `  J
) `  { y } )  /\  u  e.  ~P U ) )
10 sneq 4037 . . . . . . . . . . 11  |-  ( y  =  P  ->  { y }  =  { P } )
1110fveq2d 5870 . . . . . . . . . 10  |-  ( y  =  P  ->  (
( nei `  J
) `  { y } )  =  ( ( nei `  J
) `  { P } ) )
1211eleq2d 2537 . . . . . . . . 9  |-  ( y  =  P  ->  (
u  e.  ( ( nei `  J ) `
 { y } )  <->  u  e.  (
( nei `  J
) `  { P } ) ) )
13 selpw 4017 . . . . . . . . . 10  |-  ( u  e.  ~P U  <->  u  C_  U
)
1413a1i 11 . . . . . . . . 9  |-  ( y  =  P  ->  (
u  e.  ~P U  <->  u 
C_  U ) )
1512, 14anbi12d 710 . . . . . . . 8  |-  ( y  =  P  ->  (
( u  e.  ( ( nei `  J
) `  { y } )  /\  u  e.  ~P U )  <->  ( u  e.  ( ( nei `  J
) `  { P } )  /\  u  C_  U ) ) )
169, 15syl5bb 257 . . . . . . 7  |-  ( y  =  P  ->  (
u  e.  ( ( ( nei `  J
) `  { y } )  i^i  ~P U )  <->  ( u  e.  ( ( nei `  J
) `  { P } )  /\  u  C_  U ) ) )
1716anbi1d 704 . . . . . 6  |-  ( 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 649 . . . . . 6  |-  ( ( ( 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 261 . . . . 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
) ) ) )
2019rexbidv2 2969 . . . 4  |-  ( 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 3213 . . 3  |-  ( ( 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, 21sylan 471 . 2  |-  ( ( ( J  e. 𝑛Locally  A  /\  U  e.  J )  /\  P  e.  U
)  ->  E. u  e.  ( ( nei `  J
) `  { P } ) ( u 
C_  U  /\  ( Jt  u )  e.  A
) )
23223impa 1191 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   {csn 4027   ` cfv 5588  (class class class)co 6285   ↾t crest 14679   Topctop 19201   neicnei 19404  𝑛Locally cnlly 19772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-ov 6288  df-nlly 19774
This theorem is referenced by:  nlly2i  19783  llycmpkgen  19880
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