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Theorem eltopspOLD 18521
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 18523, 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 4291 . . . 4  |-  ( U. J TopSpOLD J  <->  <. U. J ,  J >.  e.  TopSpOLD )
2 relopab 4964 . . . . . 6  |-  Rel  { <. x ,  y >.  |  ( y  e. 
Top  /\  x  =  U. y ) }
3 df-topspOLD 18502 . . . . . . 7  |-  TopSpOLD  =  { <. x ,  y
>.  |  ( y  e.  Top  /\  x  = 
U. y ) }
43releqi 4921 . . . . . 6  |-  ( Rel  TopSpOLD  <->  Rel  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) } )
52, 4mpbir 209 . . . . 5  |-  Rel  TopSpOLD
65brrelexi 4877 . . . 4  |-  ( U. J TopSpOLD J  ->  U. J  e.  _V )
71, 6sylbir 213 . . 3  |-  ( <. U. J ,  J >.  e. 
TopSpOLD  ->  U. J  e. 
_V )
8 uniexb 6384 . . . 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 6375 . . 3  |-  ( J  e.  Top  ->  U. J  e.  _V )
12 elex 2979 . . 3  |-  ( J  e.  Top  ->  J  e.  _V )
1311, 12jca 532 . 2  |-  ( J  e.  Top  ->  ( U. J  e.  _V  /\  J  e.  _V )
)
14 eqeq1 2447 . . . . 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 2501 . . . . 5  |-  ( y  =  J  ->  (
y  e.  Top  <->  J  e.  Top ) )
17 unieq 4097 . . . . . 6  |-  ( y  =  J  ->  U. y  =  U. J )
1817eqeq2d 2452 . . . . 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 4605 . . 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 2505 . . 3  |-  ( <. U. J ,  J >.  e. 
TopSpOLD  <->  <. U. J ,  J >.  e.  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) } )
22 eqid 2441 . . . 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 1369    e. wcel 1756   _Vcvv 2970   <.cop 3881   U.cuni 4089   class class class wbr 4290   {copab 4347   Rel wrel 4843   Topctop 18496   TopSpOLDctpsOLD 18498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-xp 4844  df-rel 4845  df-topspOLD 18502
This theorem is referenced by: (None)
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