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Theorem eltopspOLD 19545
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 19547, 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. 
TopSpOLD  <->  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 4457 . . . 4  |-  ( U. J TopSpOLD J  <->  <. U. J ,  J >.  e.  TopSpOLD )
2 relopab 5138 . . . . . 6  |-  Rel  { <. x ,  y >.  |  ( y  e. 
Top  /\  x  =  U. y ) }
3 df-topspOLD 19526 . . . . . . 7  |-  TopSpOLD  =  { <. x ,  y
>.  |  ( y  e.  Top  /\  x  = 
U. y ) }
43releqi 5095 . . . . . 6  |-  ( Rel  TopSpOLD  <->  Rel  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) } )
52, 4mpbir 209 . . . . 5  |-  Rel  TopSpOLD
65brrelexi 5049 . . . 4  |-  ( U. J TopSpOLD J  ->  U. J  e.  _V )
71, 6sylbir 213 . . 3  |-  ( <. U. J ,  J >.  e. 
TopSpOLD  ->  U. J  e. 
_V )
8 uniexb 6609 . . . 4  |-  ( J  e.  _V  <->  U. J  e. 
_V )
97, 8sylibr 212 . . 3  |-  ( <. U. J ,  J >.  e. 
TopSpOLD  ->  J  e.  _V )
107, 9jca 532 . 2  |-  ( <. U. J ,  J >.  e. 
TopSpOLD  ->  ( U. J  e.  _V  /\  J  e.  _V ) )
11 uniexg 6596 . . 3  |-  ( J  e.  Top  ->  U. J  e.  _V )
12 elex 3118 . . 3  |-  ( J  e.  Top  ->  J  e.  _V )
1311, 12jca 532 . 2  |-  ( J  e.  Top  ->  ( U. J  e.  _V  /\  J  e.  _V )
)
14 eqeq1 2461 . . . . 5  |-  ( x  =  U. J  -> 
( x  =  U. y 
<-> 
U. J  =  U. y ) )
1514anbi2d 703 . . . 4  |-  ( x  =  U. J  -> 
( ( y  e. 
Top  /\  x  =  U. y )  <->  ( y  e.  Top  /\  U. J  =  U. y ) ) )
16 eleq1 2529 . . . . 5  |-  ( y  =  J  ->  (
y  e.  Top  <->  J  e.  Top ) )
17 unieq 4259 . . . . . 6  |-  ( y  =  J  ->  U. y  =  U. J )
1817eqeq2d 2471 . . . . 5  |-  ( y  =  J  ->  ( U. J  =  U. y 
<-> 
U. J  =  U. J ) )
1916, 18anbi12d 710 . . . 4  |-  ( y  =  J  ->  (
( y  e.  Top  /\ 
U. J  =  U. y )  <->  ( J  e.  Top  /\  U. J  =  U. J ) ) )
2015, 19opelopabg 4774 . . 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 2535 . . 3  |-  ( <. U. J ,  J >.  e. 
TopSpOLD  <->  <. U. J ,  J >.  e.  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) } )
22 eqid 2457 . . . 4  |-  U. J  =  U. J
2322biantru 505 . . 3  |-  ( J  e.  Top  <->  ( J  e.  Top  /\  U. J  =  U. J ) )
2420, 21, 233bitr4g 288 . 2  |-  ( ( U. J  e.  _V  /\  J  e.  _V )  ->  ( <. U. J ,  J >.  e.  TopSpOLD  <->  J  e.  Top ) )
2510, 13, 24pm5.21nii 353 1  |-  ( <. U. J ,  J >.  e. 
TopSpOLD  <->  J  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109   <.cop 4038   U.cuni 4251   class class class wbr 4456   {copab 4514   Rel wrel 5013   Topctop 19520   TopSpOLDctpsOLD 19522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-topspOLD 19526
This theorem is referenced by: (None)
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