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Theorem eltopspOLD 16756
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 16758, 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 4103 . . . 4  |-  ( U. J TopSp OLD J  <->  <. U. J ,  J >.  e.  TopSp OLD )
2 relopab 4891 . . . . . 6  |-  Rel  { <. x ,  y >.  |  ( y  e. 
Top  /\  x  =  U. y ) }
3 df-topspOLD 16737 . . . . . . 7  |-  TopSp OLD  =  { <. x ,  y
>.  |  ( y  e.  Top  /\  x  = 
U. y ) }
43releqi 4851 . . . . . 6  |-  ( Rel  TopSp OLD  <->  Rel  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) } )
52, 4mpbir 200 . . . . 5  |-  Rel  TopSp OLD
65brrelexi 4808 . . . 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 4642 . . . 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 4596 . . 3  |-  ( J  e.  Top  ->  U. J  e.  _V )
12 elex 2872 . . 3  |-  ( J  e.  Top  ->  J  e.  _V )
1311, 12jca 518 . 2  |-  ( J  e.  Top  ->  ( U. J  e.  _V  /\  J  e.  _V )
)
14 eqeq1 2364 . . . . 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 2418 . . . . 5  |-  ( y  =  J  ->  (
y  e.  Top  <->  J  e.  Top ) )
17 unieq 3915 . . . . . 6  |-  ( y  =  J  ->  U. y  =  U. J )
1817eqeq2d 2369 . . . . 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 4362 . . 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 2422 . . 3  |-  ( <. U. J ,  J >.  e. 
TopSp OLD  <->  <. U. J ,  J >.  e.  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) } )
22 eqid 2358 . . . 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 1642    e. wcel 1710   _Vcvv 2864   <.cop 3719   U.cuni 3906   class class class wbr 4102   {copab 4155   Rel wrel 4773   Topctop 16731   TopSp OLDctpsOLD 16733
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-xp 4774  df-rel 4775  df-topspOLD 16737
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