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Theorem istpsOLD 16977
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. 
TopSp OLD  <->  ( 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 16976 . 2  |-  ( <. A ,  J >.  e. 
TopSp OLD  ->  ( A  e.  _V  /\  J  e. 
_V ) )
2 simpr 448 . . . 4  |-  ( ( J  e.  Top  /\  A  =  U. J )  ->  A  =  U. J )
3 uniexg 4698 . . . . 5  |-  ( J  e.  Top  ->  U. J  e.  _V )
43adantr 452 . . . 4  |-  ( ( J  e.  Top  /\  A  =  U. J )  ->  U. J  e.  _V )
52, 4eqeltrd 2509 . . 3  |-  ( ( J  e.  Top  /\  A  =  U. J )  ->  A  e.  _V )
6 elex 2956 . . . 4  |-  ( J  e.  Top  ->  J  e.  _V )
76adantr 452 . . 3  |-  ( ( J  e.  Top  /\  A  =  U. J )  ->  J  e.  _V )
85, 7jca 519 . 2  |-  ( ( J  e.  Top  /\  A  =  U. J )  ->  ( A  e. 
_V  /\  J  e.  _V ) )
9 df-topspOLD 16956 . . . 4  |-  TopSp OLD  =  { <. x ,  y
>.  |  ( y  e.  Top  /\  x  = 
U. y ) }
109eleq2i 2499 . . 3  |-  ( <. A ,  J >.  e. 
TopSp OLD  <->  <. A ,  J >.  e.  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y
) } )
11 eqeq1 2441 . . . . 5  |-  ( x  =  A  ->  (
x  =  U. y  <->  A  =  U. y ) )
1211anbi2d 685 . . . 4  |-  ( x  =  A  ->  (
( y  e.  Top  /\  x  =  U. y
)  <->  ( y  e. 
Top  /\  A  =  U. y ) ) )
13 eleq1 2495 . . . . 5  |-  ( y  =  J  ->  (
y  e.  Top  <->  J  e.  Top ) )
14 unieq 4016 . . . . . 6  |-  ( y  =  J  ->  U. y  =  U. J )
1514eqeq2d 2446 . . . . 5  |-  ( y  =  J  ->  ( A  =  U. y  <->  A  =  U. J ) )
1613, 15anbi12d 692 . . . 4  |-  ( y  =  J  ->  (
( y  e.  Top  /\  A  =  U. y
)  <->  ( J  e. 
Top  /\  A  =  U. J ) ) )
1712, 16opelopabg 4465 . . 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 249 . 2  |-  ( ( A  e.  _V  /\  J  e.  _V )  ->  ( <. A ,  J >.  e.  TopSp OLD  <->  ( J  e. 
Top  /\  A  =  U. J ) ) )
191, 8, 18pm5.21nii 343 1  |-  ( <. A ,  J >.  e. 
TopSp OLD  <->  ( J  e. 
Top  /\  A  =  U. J ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   U.cuni 4007   {copab 4257   Topctop 16950   TopSp OLDctpsOLD 16952
This theorem is referenced by:  istps2OLD  16978  retpsOLD  18790
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-topspOLD 16956
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