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Theorem isnlly 19078
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 5696 . . . . . . 7  |-  ( j  =  J  ->  ( nei `  j )  =  ( nei `  J
) )
21fveq1d 5698 . . . . . 6  |-  ( j  =  J  ->  (
( nei `  j
) `  { y } )  =  ( ( nei `  J
) `  { y } ) )
32ineq1d 3556 . . . . 5  |-  ( j  =  J  ->  (
( ( nei `  j
) `  { y } )  i^i  ~P x )  =  ( ( ( nei `  J
) `  { y } )  i^i  ~P x ) )
4 oveq1 6103 . . . . . 6  |-  ( j  =  J  ->  (
jt  u )  =  ( Jt  u ) )
54eleq1d 2509 . . . . 5  |-  ( j  =  J  ->  (
( jt  u )  e.  A  <->  ( Jt  u )  e.  A
) )
63, 5rexeqbidv 2937 . . . 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 2740 . . 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 2930 . 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 19076 . 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 3124 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721    i^i cin 3332   ~Pcpw 3865   {csn 3882   ` cfv 5423  (class class class)co 6096   ↾t crest 14364   Topctop 18503   neicnei 18706  𝑛Locally cnlly 19074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-iota 5386  df-fv 5431  df-ov 6099  df-nlly 19076
This theorem is referenced by:  nllytop  19082  nllyi  19084  llynlly  19086  nllyss  19089  nllyrest  19095  nllyidm  19098  hausllycmp  19103  cldllycmp  19104  txnlly  19215  cnllycmp  20533
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