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Theorem istpsOLD 19547
Description: Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
istpsOLD  |-  ( <. A ,  J >.  e. 
TopSpOLD  <->  ( J  e. 
Top  /\  A  =  U. J ) )

Proof of Theorem istpsOLD
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tpsexOLD 19546 . 2  |-  ( <. A ,  J >.  e. 
TopSpOLD  ->  ( A  e.  _V  /\  J  e. 
_V ) )
2 simpr 461 . . . 4  |-  ( ( J  e.  Top  /\  A  =  U. J )  ->  A  =  U. J )
3 uniexg 6596 . . . . 5  |-  ( J  e.  Top  ->  U. J  e.  _V )
43adantr 465 . . . 4  |-  ( ( J  e.  Top  /\  A  =  U. J )  ->  U. J  e.  _V )
52, 4eqeltrd 2545 . . 3  |-  ( ( J  e.  Top  /\  A  =  U. J )  ->  A  e.  _V )
6 elex 3118 . . . 4  |-  ( J  e.  Top  ->  J  e.  _V )
76adantr 465 . . 3  |-  ( ( J  e.  Top  /\  A  =  U. J )  ->  J  e.  _V )
85, 7jca 532 . 2  |-  ( ( J  e.  Top  /\  A  =  U. J )  ->  ( A  e. 
_V  /\  J  e.  _V ) )
9 df-topspOLD 19526 . . . 4  |-  TopSpOLD  =  { <. x ,  y
>.  |  ( y  e.  Top  /\  x  = 
U. y ) }
109eleq2i 2535 . . 3  |-  ( <. A ,  J >.  e. 
TopSpOLD  <->  <. A ,  J >.  e.  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) } )
11 eqeq1 2461 . . . . 5  |-  ( x  =  A  ->  (
x  =  U. y  <->  A  =  U. y ) )
1211anbi2d 703 . . . 4  |-  ( x  =  A  ->  (
( y  e.  Top  /\  x  =  U. y
)  <->  ( y  e. 
Top  /\  A  =  U. y ) ) )
13 eleq1 2529 . . . . 5  |-  ( y  =  J  ->  (
y  e.  Top  <->  J  e.  Top ) )
14 unieq 4259 . . . . . 6  |-  ( y  =  J  ->  U. y  =  U. J )
1514eqeq2d 2471 . . . . 5  |-  ( y  =  J  ->  ( A  =  U. y  <->  A  =  U. J ) )
1613, 15anbi12d 710 . . . 4  |-  ( y  =  J  ->  (
( y  e.  Top  /\  A  =  U. y
)  <->  ( J  e. 
Top  /\  A  =  U. J ) ) )
1712, 16opelopabg 4774 . . 3  |-  ( ( A  e.  _V  /\  J  e.  _V )  ->  ( <. A ,  J >.  e.  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) }  <->  ( J  e.  Top  /\  A  = 
U. J ) ) )
1810, 17syl5bb 257 . 2  |-  ( ( A  e.  _V  /\  J  e.  _V )  ->  ( <. A ,  J >.  e.  TopSpOLD  <->  ( J  e. 
Top  /\  A  =  U. J ) ) )
191, 8, 18pm5.21nii 353 1  |-  ( <. A ,  J >.  e. 
TopSpOLD  <->  ( J  e. 
Top  /\  A  =  U. J ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109   <.cop 4038   U.cuni 4251   {copab 4514   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-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-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:  istps2OLD  19548  retpsOLD  21396
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