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Theorem isnlly 17446
Description: The property of being an n-locally  A topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
isnlly  |-  ( 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
) )
Distinct variable groups:    x, u, y, A    u, J, x, y

Proof of Theorem isnlly
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 fveq2 5661 . . . . . . 7  |-  ( j  =  J  ->  ( nei `  j )  =  ( nei `  J
) )
21fveq1d 5663 . . . . . 6  |-  ( j  =  J  ->  (
( nei `  j
) `  { y } )  =  ( ( nei `  J
) `  { y } ) )
32ineq1d 3477 . . . . 5  |-  ( j  =  J  ->  (
( ( nei `  j
) `  { y } )  i^i  ~P x )  =  ( ( ( nei `  J
) `  { y } )  i^i  ~P x ) )
4 oveq1 6020 . . . . . 6  |-  ( j  =  J  ->  (
jt  u )  =  ( Jt  u ) )
54eleq1d 2446 . . . . 5  |-  ( j  =  J  ->  (
( jt  u )  e.  A  <->  ( Jt  u )  e.  A
) )
63, 5rexeqbidv 2853 . . . 4  |-  ( j  =  J  ->  ( E. u  e.  (
( ( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  A  <->  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P x ) ( Jt  u )  e.  A
) )
76ralbidv 2662 . . 3  |-  ( j  =  J  ->  ( A. y  e.  x  E. u  e.  (
( ( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  A  <->  A. y  e.  x  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P x ) ( Jt  u )  e.  A
) )
87raleqbi1dv 2848 . 2  |-  ( j  =  J  ->  ( A. x  e.  j  A. y  e.  x  E. u  e.  (
( ( nei `  j
) `  { y } )  i^i  ~P x ) ( jt  u )  e.  A  <->  A. x  e.  J  A. y  e.  x  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P x ) ( Jt  u )  e.  A
) )
9 df-nlly 17444 . 2  |- 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 }
108, 9elrab2 3030 1  |-  ( 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
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642   E.wrex 2643    i^i cin 3255   ~Pcpw 3735   {csn 3750   ` cfv 5387  (class class class)co 6013   ↾t crest 13568   Topctop 16874   neicnei 17077  𝑛Locally cnlly 17442
This theorem is referenced by:  nllytop  17450  nllyi  17452  llynlly  17454  nllyss  17457  nllyrest  17463  nllyidm  17466  hausllycmp  17471  cldllycmp  17472  txnlly  17583  cnllycmp  18845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-iota 5351  df-fv 5395  df-ov 6016  df-nlly 17444
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