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Theorem istpsOLD 16658
Description: Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
istpsOLD  |-  ( <. A ,  J >.  e. 
TopSp OLD  <->  ( J  e. 
Top  /\  A  =  U. J ) )

Proof of Theorem istpsOLD
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tpsexOLD 16657 . 2  |-  ( <. A ,  J >.  e. 
TopSp OLD  ->  ( A  e.  _V  /\  J  e. 
_V ) )
2 simpr 447 . . . 4  |-  ( ( J  e.  Top  /\  A  =  U. J )  ->  A  =  U. J )
3 uniexg 4517 . . . . 5  |-  ( J  e.  Top  ->  U. J  e.  _V )
43adantr 451 . . . 4  |-  ( ( J  e.  Top  /\  A  =  U. J )  ->  U. J  e.  _V )
52, 4eqeltrd 2357 . . 3  |-  ( ( J  e.  Top  /\  A  =  U. J )  ->  A  e.  _V )
6 elex 2796 . . . 4  |-  ( J  e.  Top  ->  J  e.  _V )
76adantr 451 . . 3  |-  ( ( J  e.  Top  /\  A  =  U. J )  ->  J  e.  _V )
85, 7jca 518 . 2  |-  ( ( J  e.  Top  /\  A  =  U. J )  ->  ( A  e. 
_V  /\  J  e.  _V ) )
9 df-topspOLD 16637 . . . 4  |-  TopSp OLD  =  { <. x ,  y
>.  |  ( y  e.  Top  /\  x  = 
U. y ) }
109eleq2i 2347 . . 3  |-  ( <. A ,  J >.  e. 
TopSp OLD  <->  <. A ,  J >.  e.  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) } )
11 eqeq1 2289 . . . . 5  |-  ( x  =  A  ->  (
x  =  U. y  <->  A  =  U. y ) )
1211anbi2d 684 . . . 4  |-  ( x  =  A  ->  (
( y  e.  Top  /\  x  =  U. y
)  <->  ( y  e. 
Top  /\  A  =  U. y ) ) )
13 eleq1 2343 . . . . 5  |-  ( y  =  J  ->  (
y  e.  Top  <->  J  e.  Top ) )
14 unieq 3836 . . . . . 6  |-  ( y  =  J  ->  U. y  =  U. J )
1514eqeq2d 2294 . . . . 5  |-  ( y  =  J  ->  ( A  =  U. y  <->  A  =  U. J ) )
1613, 15anbi12d 691 . . . 4  |-  ( y  =  J  ->  (
( y  e.  Top  /\  A  =  U. y
)  <->  ( J  e. 
Top  /\  A  =  U. J ) ) )
1712, 16opelopabg 4283 . . 3  |-  ( ( A  e.  _V  /\  J  e.  _V )  ->  ( <. A ,  J >.  e.  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) }  <->  ( J  e.  Top  /\  A  = 
U. J ) ) )
1810, 17syl5bb 248 . 2  |-  ( ( A  e.  _V  /\  J  e.  _V )  ->  ( <. A ,  J >.  e.  TopSp OLD  <->  ( J  e. 
Top  /\  A  =  U. J ) ) )
191, 8, 18pm5.21nii 342 1  |-  ( <. A ,  J >.  e. 
TopSp OLD  <->  ( J  e. 
Top  /\  A  =  U. J ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   U.cuni 3827   {copab 4076   Topctop 16631   TopSp OLDctpsOLD 16633
This theorem is referenced by:  istps2OLD  16659  retpsOLD  18273
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  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-opab 4078  df-xp 4695  df-rel 4696  df-topspOLD 16637
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