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Theorem isnlly 17211
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 5541 . . . . . . 7  |-  ( j  =  J  ->  ( nei `  j )  =  ( nei `  J
) )
21fveq1d 5543 . . . . . 6  |-  ( j  =  J  ->  (
( nei `  j
) `  { y } )  =  ( ( nei `  J
) `  { y } ) )
32ineq1d 3382 . . . . 5  |-  ( j  =  J  ->  (
( ( nei `  j
) `  { y } )  i^i  ~P x )  =  ( ( ( nei `  J
) `  { y } )  i^i  ~P x ) )
4 oveq1 5881 . . . . . 6  |-  ( j  =  J  ->  (
jt  u )  =  ( Jt  u ) )
54eleq1d 2362 . . . . 5  |-  ( j  =  J  ->  (
( jt  u )  e.  A  <->  ( Jt  u )  e.  A
) )
63, 5rexeqbidv 2762 . . . 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 2576 . . 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 2757 . 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 17209 . 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 2938 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164   ~Pcpw 3638   {csn 3653   ` cfv 5271  (class class class)co 5874   ↾t crest 13341   Topctop 16647   neicnei 16850  𝑛Locally cnlly 17207
This theorem is referenced by:  nllytop  17215  nllyi  17217  llynlly  17219  nllyss  17222  nllyrest  17228  nllyidm  17231  hausllycmp  17236  cldllycmp  17237  txnlly  17347  cnllycmp  18470
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-nlly 17209
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