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Theorem isnrm 17063
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 5525 . . . . 5  |-  ( j  =  J  ->  ( Clsd `  j )  =  ( Clsd `  J
) )
21ineq1d 3369 . . . 4  |-  ( j  =  J  ->  (
( Clsd `  j )  i^i  ~P x )  =  ( ( Clsd `  J
)  i^i  ~P x
) )
3 fveq2 5525 . . . . . . . 8  |-  ( j  =  J  ->  ( cls `  j )  =  ( cls `  J
) )
43fveq1d 5527 . . . . . . 7  |-  ( j  =  J  ->  (
( cls `  j
) `  z )  =  ( ( cls `  J ) `  z
) )
54sseq1d 3205 . . . . . 6  |-  ( j  =  J  ->  (
( ( cls `  j
) `  z )  C_  x  <->  ( ( cls `  J ) `  z
)  C_  x )
)
65anbi2d 684 . . . . 5  |-  ( j  =  J  ->  (
( y  C_  z  /\  ( ( cls `  j
) `  z )  C_  x )  <->  ( y  C_  z  /\  ( ( cls `  J ) `
 z )  C_  x ) ) )
76rexeqbi1dv 2745 . . . 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 2748 . . 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 2744 . 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 17045 . 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 2925 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   ` cfv 5255   Topctop 16631   Clsdccld 16753   clsccl 16755   Nrmcnrm 17038
This theorem is referenced by:  nrmtop  17064  nrmsep3  17083  isnrm2  17086  kqnrmlem1  17434  kqnrmlem2  17435  nrmhmph  17485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-nrm 17045
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