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Theorem isnrm 17401
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 5730 . . . . 5  |-  ( j  =  J  ->  ( Clsd `  j )  =  ( Clsd `  J
) )
21ineq1d 3543 . . . 4  |-  ( j  =  J  ->  (
( Clsd `  j )  i^i  ~P x )  =  ( ( Clsd `  J
)  i^i  ~P x
) )
3 fveq2 5730 . . . . . . . 8  |-  ( j  =  J  ->  ( cls `  j )  =  ( cls `  J
) )
43fveq1d 5732 . . . . . . 7  |-  ( j  =  J  ->  (
( cls `  j
) `  z )  =  ( ( cls `  J ) `  z
) )
54sseq1d 3377 . . . . . 6  |-  ( j  =  J  ->  (
( ( cls `  j
) `  z )  C_  x  <->  ( ( cls `  J ) `  z
)  C_  x )
)
65anbi2d 686 . . . . 5  |-  ( j  =  J  ->  (
( y  C_  z  /\  ( ( cls `  j
) `  z )  C_  x )  <->  ( y  C_  z  /\  ( ( cls `  J ) `
 z )  C_  x ) ) )
76rexeqbi1dv 2915 . . . 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 2918 . . 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 2914 . 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 17383 . 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 3096 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 set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   ` cfv 5456   Topctop 16960   Clsdccld 17082   clsccl 17084   Nrmcnrm 17376
This theorem is referenced by:  nrmtop  17402  nrmsep3  17421  isnrm2  17424  kqnrmlem1  17777  kqnrmlem2  17778  nrmhmph  17828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-nrm 17383
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