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Theorem istpsOLD 16714
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 16713 . 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 4554 . . . . 5  |-  ( J  e.  Top  ->  U. J  e.  _V )
43adantr 451 . . . 4  |-  ( ( J  e.  Top  /\  A  =  U. J )  ->  U. J  e.  _V )
52, 4eqeltrd 2390 . . 3  |-  ( ( J  e.  Top  /\  A  =  U. J )  ->  A  e.  _V )
6 elex 2830 . . . 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 16693 . . . 4  |-  TopSp OLD  =  { <. x ,  y
>.  |  ( y  e.  Top  /\  x  = 
U. y ) }
109eleq2i 2380 . . 3  |-  ( <. A ,  J >.  e. 
TopSp OLD  <->  <. A ,  J >.  e.  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) } )
11 eqeq1 2322 . . . . 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 2376 . . . . 5  |-  ( y  =  J  ->  (
y  e.  Top  <->  J  e.  Top ) )
14 unieq 3873 . . . . . 6  |-  ( y  =  J  ->  U. y  =  U. J )
1514eqeq2d 2327 . . . . 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 4320 . . 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 1633    e. wcel 1701   _Vcvv 2822   <.cop 3677   U.cuni 3864   {copab 4113   Topctop 16687   TopSp OLDctpsOLD 16689
This theorem is referenced by:  istps2OLD  16715  retpsOLD  18325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-xp 4732  df-rel 4733  df-topspOLD 16693
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