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Theorem isnrm 20127
Description: The predicate "is a normal space." Much like the case for regular spaces, normal does not imply Hausdorff or even regular. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
isnrm  |-  ( J  e.  Nrm  <->  ( J  e.  Top  /\  A. x  e.  J  A. y  e.  ( ( Clsd `  J
)  i^i  ~P x
) E. z  e.  J  ( y  C_  z  /\  ( ( cls `  J ) `  z
)  C_  x )
) )
Distinct variable group:    x, y, z, J

Proof of Theorem isnrm
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 fveq2 5848 . . . . 5  |-  ( j  =  J  ->  ( Clsd `  j )  =  ( Clsd `  J
) )
21ineq1d 3639 . . . 4  |-  ( j  =  J  ->  (
( Clsd `  j )  i^i  ~P x )  =  ( ( Clsd `  J
)  i^i  ~P x
) )
3 fveq2 5848 . . . . . . . 8  |-  ( j  =  J  ->  ( cls `  j )  =  ( cls `  J
) )
43fveq1d 5850 . . . . . . 7  |-  ( j  =  J  ->  (
( cls `  j
) `  z )  =  ( ( cls `  J ) `  z
) )
54sseq1d 3468 . . . . . 6  |-  ( j  =  J  ->  (
( ( cls `  j
) `  z )  C_  x  <->  ( ( cls `  J ) `  z
)  C_  x )
)
65anbi2d 702 . . . . 5  |-  ( j  =  J  ->  (
( y  C_  z  /\  ( ( cls `  j
) `  z )  C_  x )  <->  ( y  C_  z  /\  ( ( cls `  J ) `
 z )  C_  x ) ) )
76rexeqbi1dv 3012 . . . 4  |-  ( j  =  J  ->  ( E. z  e.  j 
( y  C_  z  /\  ( ( cls `  j
) `  z )  C_  x )  <->  E. z  e.  J  ( y  C_  z  /\  ( ( cls `  J ) `
 z )  C_  x ) ) )
82, 7raleqbidv 3017 . . 3  |-  ( j  =  J  ->  ( A. y  e.  (
( Clsd `  j )  i^i  ~P x ) E. z  e.  j  ( y  C_  z  /\  ( ( cls `  j
) `  z )  C_  x )  <->  A. y  e.  ( ( Clsd `  J
)  i^i  ~P x
) E. z  e.  J  ( y  C_  z  /\  ( ( cls `  J ) `  z
)  C_  x )
) )
98raleqbi1dv 3011 . 2  |-  ( j  =  J  ->  ( A. x  e.  j  A. y  e.  (
( Clsd `  j )  i^i  ~P x ) E. z  e.  j  ( y  C_  z  /\  ( ( cls `  j
) `  z )  C_  x )  <->  A. x  e.  J  A. y  e.  ( ( Clsd `  J
)  i^i  ~P x
) E. z  e.  J  ( y  C_  z  /\  ( ( cls `  J ) `  z
)  C_  x )
) )
10 df-nrm 20109 . 2  |-  Nrm  =  { j  e.  Top  | 
A. x  e.  j 
A. y  e.  ( ( Clsd `  j
)  i^i  ~P x
) E. z  e.  j  ( y  C_  z  /\  ( ( cls `  j ) `  z
)  C_  x ) }
119, 10elrab2 3208 1  |-  ( J  e.  Nrm  <->  ( J  e.  Top  /\  A. x  e.  J  A. y  e.  ( ( Clsd `  J
)  i^i  ~P x
) E. z  e.  J  ( y  C_  z  /\  ( ( cls `  J ) `  z
)  C_  x )
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   E.wrex 2754    i^i cin 3412    C_ wss 3413   ~Pcpw 3954   ` cfv 5568   Topctop 19684   Clsdccld 19807   clsccl 19809   Nrmcnrm 20102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5532  df-fv 5576  df-nrm 20109
This theorem is referenced by:  nrmtop  20128  nrmsep3  20147  isnrm2  20150  kqnrmlem1  20534  kqnrmlem2  20535  nrmhmph  20585
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