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Theorem indistps2ALT 18736
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 18734 from the structural version indistps 18733. (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 5801 . . . 4  |-  ( Base `  K )  e.  _V
31, 2eqeltrri 2536 . . 3  |-  A  e. 
_V
4 indistopon 18723 . . 3  |-  ( A  e.  _V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
53, 4ax-mp 5 . 2  |-  { (/) ,  A }  e.  (TopOn `  A )
61eqcomi 2464 . . 3  |-  A  =  ( Base `  K
)
7 indistps2ALT.j . . . 4  |-  ( TopOpen `  K )  =  { (/)
,  A }
87eqcomi 2464 . . 3  |-  { (/) ,  A }  =  (
TopOpen `  K )
96, 8istps 18659 . 2  |-  ( K  e.  TopSp 
<->  { (/) ,  A }  e.  (TopOn `  A )
)
105, 9mpbir 209 1  |-  K  e. 
TopSp
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   _Vcvv 3070   (/)c0 3737   {cpr 3979   ` cfv 5518   Basecbs 14278   TopOpenctopn 14464  TopOnctopon 18617   TopSpctps 18619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5481  df-fun 5520  df-fv 5526  df-top 18621  df-topon 18624  df-topsp 18625
This theorem is referenced by: (None)
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