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Theorem isreg 17076
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 5541 . . . . . . . 8  |-  ( j  =  J  ->  ( cls `  j )  =  ( cls `  J
) )
21fveq1d 5543 . . . . . . 7  |-  ( j  =  J  ->  (
( cls `  j
) `  z )  =  ( ( cls `  J ) `  z
) )
32sseq1d 3218 . . . . . 6  |-  ( j  =  J  ->  (
( ( cls `  j
) `  z )  C_  x  <->  ( ( cls `  J ) `  z
)  C_  x )
)
43anbi2d 684 . . . . 5  |-  ( j  =  J  ->  (
( y  e.  z  /\  ( ( cls `  j ) `  z
)  C_  x )  <->  ( y  e.  z  /\  ( ( cls `  J
) `  z )  C_  x ) ) )
54rexeqbi1dv 2758 . . . 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 2576 . . 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 2757 . 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 17060 . 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 2938 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   ` cfv 5271   Topctop 16647   clsccl 16771   Regcreg 17053
This theorem is referenced by:  regtop  17077  regsep  17078  isreg2  17121  kqreglem1  17448  kqreglem2  17449  nrmr0reg  17456  reghmph  17500
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-reg 17060
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