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Theorem indistps2ALT 19281
Description: The indiscrete topology on a set  A expressed as a topological space, using direct component assignments. Here we show how to derive the direct component assignment version indistps2 19279 from the structural version indistps 19278. (Contributed by NM, 24-Oct-2012.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
indistps2ALT.a  |-  ( Base `  K )  =  A
indistps2ALT.j  |-  ( TopOpen `  K )  =  { (/)
,  A }
Assertion
Ref Expression
indistps2ALT  |-  K  e. 
TopSp

Proof of Theorem indistps2ALT
StepHypRef Expression
1 indistps2ALT.a . . . 4  |-  ( Base `  K )  =  A
2 fvex 5874 . . . 4  |-  ( Base `  K )  e.  _V
31, 2eqeltrri 2552 . . 3  |-  A  e. 
_V
4 indistopon 19268 . . 3  |-  ( A  e.  _V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
53, 4ax-mp 5 . 2  |-  { (/) ,  A }  e.  (TopOn `  A )
61eqcomi 2480 . . 3  |-  A  =  ( Base `  K
)
7 indistps2ALT.j . . . 4  |-  ( TopOpen `  K )  =  { (/)
,  A }
87eqcomi 2480 . . 3  |-  { (/) ,  A }  =  (
TopOpen `  K )
96, 8istps 19204 . 2  |-  ( K  e.  TopSp 
<->  { (/) ,  A }  e.  (TopOn `  A )
)
105, 9mpbir 209 1  |-  K  e. 
TopSp
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785   {cpr 4029   ` cfv 5586   Basecbs 14486   TopOpenctopn 14673  TopOnctopon 19162   TopSpctps 19164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-top 19166  df-topon 19169  df-topsp 19170
This theorem is referenced by: (None)
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