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Theorem isreg 20290
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 5825 . . . . . . . 8  |-  ( j  =  J  ->  ( cls `  j )  =  ( cls `  J
) )
21fveq1d 5827 . . . . . . 7  |-  ( j  =  J  ->  (
( cls `  j
) `  z )  =  ( ( cls `  J ) `  z
) )
32sseq1d 3434 . . . . . 6  |-  ( j  =  J  ->  (
( ( cls `  j
) `  z )  C_  x  <->  ( ( cls `  J ) `  z
)  C_  x )
)
43anbi2d 708 . . . . 5  |-  ( j  =  J  ->  (
( y  e.  z  /\  ( ( cls `  j ) `  z
)  C_  x )  <->  ( y  e.  z  /\  ( ( cls `  J
) `  z )  C_  x ) ) )
54rexeqbi1dv 2973 . . . 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 2804 . . 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 2972 . 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 20274 . 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 3173 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 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2714   E.wrex 2715    C_ wss 3379   ` cfv 5544   Topctop 19859   clsccl 19975   Regcreg 20267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-iota 5508  df-fv 5552  df-reg 20274
This theorem is referenced by:  regtop  20291  regsep  20292  isreg2  20335  kqreglem1  20698  kqreglem2  20699  nrmr0reg  20706  reghmph  20750  utopreg  21209
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