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Theorem isnlly 17195
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 5525 . . . . . . 7  |-  ( j  =  J  ->  ( nei `  j )  =  ( nei `  J
) )
21fveq1d 5527 . . . . . 6  |-  ( j  =  J  ->  (
( nei `  j
) `  { y } )  =  ( ( nei `  J
) `  { y } ) )
32ineq1d 3369 . . . . 5  |-  ( j  =  J  ->  (
( ( nei `  j
) `  { y } )  i^i  ~P x )  =  ( ( ( nei `  J
) `  { y } )  i^i  ~P x ) )
4 oveq1 5865 . . . . . 6  |-  ( j  =  J  ->  (
jt  u )  =  ( Jt  u ) )
54eleq1d 2349 . . . . 5  |-  ( j  =  J  ->  (
( jt  u )  e.  A  <->  ( Jt  u )  e.  A
) )
63, 5rexeqbidv 2749 . . . 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 2563 . . 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 2744 . 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 17193 . 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 2925 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151   ~Pcpw 3625   {csn 3640   ` cfv 5255  (class class class)co 5858   ↾t crest 13325   Topctop 16631   neicnei 16834  𝑛Locally cnlly 17191
This theorem is referenced by:  nllytop  17199  nllyi  17201  llynlly  17203  nllyss  17206  nllyrest  17212  nllyidm  17215  hausllycmp  17220  cldllycmp  17221  txnlly  17331  cnllycmp  18454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-nlly 17193
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