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Theorem indistps2ALT 17070
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 17068 from the structural version indistps 17067. (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 5734 . . . 4  |-  ( Base `  K )  e.  _V
31, 2eqeltrri 2506 . . 3  |-  A  e. 
_V
4 indistopon 17057 . . 3  |-  ( A  e.  _V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
53, 4ax-mp 8 . 2  |-  { (/) ,  A }  e.  (TopOn `  A )
61eqcomi 2439 . . 3  |-  A  =  ( Base `  K
)
7 indistps2ALT.j . . . 4  |-  ( TopOpen `  K )  =  { (/)
,  A }
87eqcomi 2439 . . 3  |-  { (/) ,  A }  =  (
TopOpen `  K )
96, 8istps 16993 . 2  |-  ( K  e.  TopSp 
<->  { (/) ,  A }  e.  (TopOn `  A )
)
105, 9mpbir 201 1  |-  K  e. 
TopSp
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2948   (/)c0 3620   {cpr 3807   ` cfv 5446   Basecbs 13461   TopOpenctopn 13641  TopOnctopon 16951   TopSpctps 16953
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-pow 4369  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-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-top 16955  df-topon 16958  df-topsp 16959
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