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Theorem isreg 17396
Description: The predicate "is a regular space." In a regular space, any open neighborhood has a closed subneighborhood. Note that some authors require the space to be Hausdorff (which would make it the same as T3), but we reserve the phrase "regular Hausdorff" for that as many topologists do. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
isreg  |-  ( J  e.  Reg  <->  ( J  e.  Top  /\  A. x  e.  J  A. y  e.  x  E. z  e.  J  ( y  e.  z  /\  (
( cls `  J
) `  z )  C_  x ) ) )
Distinct variable group:    x, y, z, J

Proof of Theorem isreg
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 fveq2 5728 . . . . . . . 8  |-  ( j  =  J  ->  ( cls `  j )  =  ( cls `  J
) )
21fveq1d 5730 . . . . . . 7  |-  ( j  =  J  ->  (
( cls `  j
) `  z )  =  ( ( cls `  J ) `  z
) )
32sseq1d 3375 . . . . . 6  |-  ( j  =  J  ->  (
( ( cls `  j
) `  z )  C_  x  <->  ( ( cls `  J ) `  z
)  C_  x )
)
43anbi2d 685 . . . . 5  |-  ( j  =  J  ->  (
( y  e.  z  /\  ( ( cls `  j ) `  z
)  C_  x )  <->  ( y  e.  z  /\  ( ( cls `  J
) `  z )  C_  x ) ) )
54rexeqbi1dv 2913 . . . 4  |-  ( j  =  J  ->  ( E. z  e.  j 
( y  e.  z  /\  ( ( cls `  j ) `  z
)  C_  x )  <->  E. z  e.  J  ( y  e.  z  /\  ( ( cls `  J
) `  z )  C_  x ) ) )
65ralbidv 2725 . . 3  |-  ( j  =  J  ->  ( A. y  e.  x  E. z  e.  j 
( y  e.  z  /\  ( ( cls `  j ) `  z
)  C_  x )  <->  A. y  e.  x  E. z  e.  J  (
y  e.  z  /\  ( ( cls `  J
) `  z )  C_  x ) ) )
76raleqbi1dv 2912 . 2  |-  ( j  =  J  ->  ( A. x  e.  j  A. y  e.  x  E. z  e.  j 
( y  e.  z  /\  ( ( cls `  j ) `  z
)  C_  x )  <->  A. x  e.  J  A. y  e.  x  E. z  e.  J  (
y  e.  z  /\  ( ( cls `  J
) `  z )  C_  x ) ) )
8 df-reg 17380 . 2  |-  Reg  =  { j  e.  Top  | 
A. x  e.  j 
A. y  e.  x  E. z  e.  j 
( y  e.  z  /\  ( ( cls `  j ) `  z
)  C_  x ) }
97, 8elrab2 3094 1  |-  ( J  e.  Reg  <->  ( J  e.  Top  /\  A. x  e.  J  A. y  e.  x  E. z  e.  J  ( y  e.  z  /\  (
( cls `  J
) `  z )  C_  x ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    C_ wss 3320   ` cfv 5454   Topctop 16958   clsccl 17082   Regcreg 17373
This theorem is referenced by:  regtop  17397  regsep  17398  isreg2  17441  kqreglem1  17773  kqreglem2  17774  nrmr0reg  17781  reghmph  17825  utopreg  18282
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-reg 17380
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