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Theorem isreg 19615
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 5865 . . . . . . . 8  |-  ( j  =  J  ->  ( cls `  j )  =  ( cls `  J
) )
21fveq1d 5867 . . . . . . 7  |-  ( j  =  J  ->  (
( cls `  j
) `  z )  =  ( ( cls `  J ) `  z
) )
32sseq1d 3531 . . . . . 6  |-  ( j  =  J  ->  (
( ( cls `  j
) `  z )  C_  x  <->  ( ( cls `  J ) `  z
)  C_  x )
)
43anbi2d 703 . . . . 5  |-  ( j  =  J  ->  (
( y  e.  z  /\  ( ( cls `  j ) `  z
)  C_  x )  <->  ( y  e.  z  /\  ( ( cls `  J
) `  z )  C_  x ) ) )
54rexeqbi1dv 3067 . . . 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 2903 . . 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 3066 . 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 19599 . 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 3263 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   ` cfv 5587   Topctop 19177   clsccl 19301   Regcreg 19592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5550  df-fv 5595  df-reg 19599
This theorem is referenced by:  regtop  19616  regsep  19617  isreg2  19660  kqreglem1  19993  kqreglem2  19994  nrmr0reg  20001  reghmph  20045  utopreg  20506
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