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Theorem eltopspOLD 16656
Description: Construct a topological space from a topology and vice-versa. We say that  A is a topology on  U. A. (This could be proved more efficiently from istpsOLD 16658, but the proof here does not require the Axiom of Regularity.) (Contributed by NM, 8-Mar-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
eltopspOLD  |-  ( <. U. J ,  J >.  e. 
TopSp OLD  <->  J  e.  Top )

Proof of Theorem eltopspOLD
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4024 . . . 4  |-  ( U. J TopSp OLD J  <->  <. U. J ,  J >.  e.  TopSp OLD )
2 relopab 4812 . . . . . 6  |-  Rel  { <. x ,  y >.  |  ( y  e. 
Top  /\  x  =  U. y ) }
3 df-topspOLD 16637 . . . . . . 7  |-  TopSp OLD  =  { <. x ,  y
>.  |  ( y  e.  Top  /\  x  = 
U. y ) }
43releqi 4772 . . . . . 6  |-  ( Rel  TopSp OLD  <->  Rel  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) } )
52, 4mpbir 200 . . . . 5  |-  Rel  TopSp OLD
65brrelexi 4729 . . . 4  |-  ( U. J TopSp OLD J  ->  U. J  e.  _V )
71, 6sylbir 204 . . 3  |-  ( <. U. J ,  J >.  e. 
TopSp OLD  ->  U. J  e. 
_V )
8 uniexb 4563 . . . 4  |-  ( J  e.  _V  <->  U. J  e. 
_V )
97, 8sylibr 203 . . 3  |-  ( <. U. J ,  J >.  e. 
TopSp OLD  ->  J  e.  _V )
107, 9jca 518 . 2  |-  ( <. U. J ,  J >.  e. 
TopSp OLD  ->  ( U. J  e.  _V  /\  J  e.  _V ) )
11 uniexg 4517 . . 3  |-  ( J  e.  Top  ->  U. J  e.  _V )
12 elex 2796 . . 3  |-  ( J  e.  Top  ->  J  e.  _V )
1311, 12jca 518 . 2  |-  ( J  e.  Top  ->  ( U. J  e.  _V  /\  J  e.  _V )
)
14 eqeq1 2289 . . . . 5  |-  ( x  =  U. J  -> 
( x  =  U. y 
<-> 
U. J  =  U. y ) )
1514anbi2d 684 . . . 4  |-  ( x  =  U. J  -> 
( ( y  e. 
Top  /\  x  =  U. y )  <->  ( y  e.  Top  /\  U. J  =  U. y ) ) )
16 eleq1 2343 . . . . 5  |-  ( y  =  J  ->  (
y  e.  Top  <->  J  e.  Top ) )
17 unieq 3836 . . . . . 6  |-  ( y  =  J  ->  U. y  =  U. J )
1817eqeq2d 2294 . . . . 5  |-  ( y  =  J  ->  ( U. J  =  U. y 
<-> 
U. J  =  U. J ) )
1916, 18anbi12d 691 . . . 4  |-  ( y  =  J  ->  (
( y  e.  Top  /\ 
U. J  =  U. y )  <->  ( J  e.  Top  /\  U. J  =  U. J ) ) )
2015, 19opelopabg 4283 . . 3  |-  ( ( U. J  e.  _V  /\  J  e.  _V )  ->  ( <. U. J ,  J >.  e.  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) }  <->  ( J  e.  Top  /\  U. J  =  U. J ) ) )
213eleq2i 2347 . . 3  |-  ( <. U. J ,  J >.  e. 
TopSp OLD  <->  <. U. J ,  J >.  e.  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) } )
22 eqid 2283 . . . 4  |-  U. J  =  U. J
2322biantru 491 . . 3  |-  ( J  e.  Top  <->  ( J  e.  Top  /\  U. J  =  U. J ) )
2420, 21, 233bitr4g 279 . 2  |-  ( ( U. J  e.  _V  /\  J  e.  _V )  ->  ( <. U. J ,  J >.  e.  TopSp OLD  <->  J  e.  Top ) )
2510, 13, 24pm5.21nii 342 1  |-  ( <. U. J ,  J >.  e. 
TopSp OLD  <->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   U.cuni 3827   class class class wbr 4023   {copab 4076   Rel wrel 4694   Topctop 16631   TopSp OLDctpsOLD 16633
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-pow 4188  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-pw 3627  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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