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Theorem isnlly 17524
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 5720 . . . . . . 7  |-  ( j  =  J  ->  ( nei `  j )  =  ( nei `  J
) )
21fveq1d 5722 . . . . . 6  |-  ( j  =  J  ->  (
( nei `  j
) `  { y } )  =  ( ( nei `  J
) `  { y } ) )
32ineq1d 3533 . . . . 5  |-  ( j  =  J  ->  (
( ( nei `  j
) `  { y } )  i^i  ~P x )  =  ( ( ( nei `  J
) `  { y } )  i^i  ~P x ) )
4 oveq1 6080 . . . . . 6  |-  ( j  =  J  ->  (
jt  u )  =  ( Jt  u ) )
54eleq1d 2501 . . . . 5  |-  ( j  =  J  ->  (
( jt  u )  e.  A  <->  ( Jt  u )  e.  A
) )
63, 5rexeqbidv 2909 . . . 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 2717 . . 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 2904 . 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 17522 . 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 3086 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 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    i^i cin 3311   ~Pcpw 3791   {csn 3806   ` cfv 5446  (class class class)co 6073   ↾t crest 13640   Topctop 16950   neicnei 17153  𝑛Locally cnlly 17520
This theorem is referenced by:  nllytop  17528  nllyi  17530  llynlly  17532  nllyss  17535  nllyrest  17541  nllyidm  17544  hausllycmp  17549  cldllycmp  17550  txnlly  17661  cnllycmp  18973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-nlly 17522
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