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Theorem eltopspOLD 16983
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 16985, 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 4213 . . . 4  |-  ( U. J TopSp OLD J  <->  <. U. J ,  J >.  e.  TopSp OLD )
2 relopab 5001 . . . . . 6  |-  Rel  { <. x ,  y >.  |  ( y  e. 
Top  /\  x  =  U. y ) }
3 df-topspOLD 16964 . . . . . . 7  |-  TopSp OLD  =  { <. x ,  y
>.  |  ( y  e.  Top  /\  x  = 
U. y ) }
43releqi 4960 . . . . . 6  |-  ( Rel  TopSp OLD  <->  Rel  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) } )
52, 4mpbir 201 . . . . 5  |-  Rel  TopSp OLD
65brrelexi 4918 . . . 4  |-  ( U. J TopSp OLD J  ->  U. J  e.  _V )
71, 6sylbir 205 . . 3  |-  ( <. U. J ,  J >.  e. 
TopSp OLD  ->  U. J  e. 
_V )
8 uniexb 4752 . . . 4  |-  ( J  e.  _V  <->  U. J  e. 
_V )
97, 8sylibr 204 . . 3  |-  ( <. U. J ,  J >.  e. 
TopSp OLD  ->  J  e.  _V )
107, 9jca 519 . 2  |-  ( <. U. J ,  J >.  e. 
TopSp OLD  ->  ( U. J  e.  _V  /\  J  e.  _V ) )
11 uniexg 4706 . . 3  |-  ( J  e.  Top  ->  U. J  e.  _V )
12 elex 2964 . . 3  |-  ( J  e.  Top  ->  J  e.  _V )
1311, 12jca 519 . 2  |-  ( J  e.  Top  ->  ( U. J  e.  _V  /\  J  e.  _V )
)
14 eqeq1 2442 . . . . 5  |-  ( x  =  U. J  -> 
( x  =  U. y 
<-> 
U. J  =  U. y ) )
1514anbi2d 685 . . . 4  |-  ( x  =  U. J  -> 
( ( y  e. 
Top  /\  x  =  U. y )  <->  ( y  e.  Top  /\  U. J  =  U. y ) ) )
16 eleq1 2496 . . . . 5  |-  ( y  =  J  ->  (
y  e.  Top  <->  J  e.  Top ) )
17 unieq 4024 . . . . . 6  |-  ( y  =  J  ->  U. y  =  U. J )
1817eqeq2d 2447 . . . . 5  |-  ( y  =  J  ->  ( U. J  =  U. y 
<-> 
U. J  =  U. J ) )
1916, 18anbi12d 692 . . . 4  |-  ( y  =  J  ->  (
( y  e.  Top  /\ 
U. J  =  U. y )  <->  ( J  e.  Top  /\  U. J  =  U. J ) ) )
2015, 19opelopabg 4473 . . 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 2500 . . 3  |-  ( <. U. J ,  J >.  e. 
TopSp OLD  <->  <. U. J ,  J >.  e.  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) } )
22 eqid 2436 . . . 4  |-  U. J  =  U. J
2322biantru 492 . . 3  |-  ( J  e.  Top  <->  ( J  e.  Top  /\  U. J  =  U. J ) )
2420, 21, 233bitr4g 280 . 2  |-  ( ( U. J  e.  _V  /\  J  e.  _V )  ->  ( <. U. J ,  J >.  e.  TopSp OLD  <->  J  e.  Top ) )
2510, 13, 24pm5.21nii 343 1  |-  ( <. U. J ,  J >.  e. 
TopSp OLD  <->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817   U.cuni 4015   class class class wbr 4212   {copab 4265   Rel wrel 4883   Topctop 16958   TopSp OLDctpsOLD 16960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-topspOLD 16964
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