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Theorem eltopspOLD 16985
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 16987, 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 4215 . . . 4  |-  ( U. J TopSp OLD J  <->  <. U. J ,  J >.  e.  TopSp OLD )
2 relopab 5003 . . . . . 6  |-  Rel  { <. x ,  y >.  |  ( y  e. 
Top  /\  x  =  U. y ) }
3 df-topspOLD 16966 . . . . . . 7  |-  TopSp OLD  =  { <. x ,  y
>.  |  ( y  e.  Top  /\  x  = 
U. y ) }
43releqi 4962 . . . . . 6  |-  ( Rel  TopSp OLD  <->  Rel  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) } )
52, 4mpbir 202 . . . . 5  |-  Rel  TopSp OLD
65brrelexi 4920 . . . 4  |-  ( U. J TopSp OLD J  ->  U. J  e.  _V )
71, 6sylbir 206 . . 3  |-  ( <. U. J ,  J >.  e. 
TopSp OLD  ->  U. J  e. 
_V )
8 uniexb 4754 . . . 4  |-  ( J  e.  _V  <->  U. J  e. 
_V )
97, 8sylibr 205 . . 3  |-  ( <. U. J ,  J >.  e. 
TopSp OLD  ->  J  e.  _V )
107, 9jca 520 . 2  |-  ( <. U. J ,  J >.  e. 
TopSp OLD  ->  ( U. J  e.  _V  /\  J  e.  _V ) )
11 uniexg 4708 . . 3  |-  ( J  e.  Top  ->  U. J  e.  _V )
12 elex 2966 . . 3  |-  ( J  e.  Top  ->  J  e.  _V )
1311, 12jca 520 . 2  |-  ( J  e.  Top  ->  ( U. J  e.  _V  /\  J  e.  _V )
)
14 eqeq1 2444 . . . . 5  |-  ( x  =  U. J  -> 
( x  =  U. y 
<-> 
U. J  =  U. y ) )
1514anbi2d 686 . . . 4  |-  ( x  =  U. J  -> 
( ( y  e. 
Top  /\  x  =  U. y )  <->  ( y  e.  Top  /\  U. J  =  U. y ) ) )
16 eleq1 2498 . . . . 5  |-  ( y  =  J  ->  (
y  e.  Top  <->  J  e.  Top ) )
17 unieq 4026 . . . . . 6  |-  ( y  =  J  ->  U. y  =  U. J )
1817eqeq2d 2449 . . . . 5  |-  ( y  =  J  ->  ( U. J  =  U. y 
<-> 
U. J  =  U. J ) )
1916, 18anbi12d 693 . . . 4  |-  ( y  =  J  ->  (
( y  e.  Top  /\ 
U. J  =  U. y )  <->  ( J  e.  Top  /\  U. J  =  U. J ) ) )
2015, 19opelopabg 4475 . . 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 2502 . . 3  |-  ( <. U. J ,  J >.  e. 
TopSp OLD  <->  <. U. J ,  J >.  e.  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) } )
22 eqid 2438 . . . 4  |-  U. J  =  U. J
2322biantru 493 . . 3  |-  ( J  e.  Top  <->  ( J  e.  Top  /\  U. J  =  U. J ) )
2420, 21, 233bitr4g 281 . 2  |-  ( ( U. J  e.  _V  /\  J  e.  _V )  ->  ( <. U. J ,  J >.  e.  TopSp OLD  <->  J  e.  Top ) )
2510, 13, 24pm5.21nii 344 1  |-  ( <. U. J ,  J >.  e. 
TopSp OLD  <->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819   U.cuni 4017   class class class wbr 4214   {copab 4267   Rel wrel 4885   Topctop 16960   TopSp OLDctpsOLD 16962
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-topspOLD 16966
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