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Theorem indistps2ALT 16767
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 16765 from the structural version indistps 16764. (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 5555 . . . 4  |-  ( Base `  K )  e.  _V
31, 2eqeltrri 2367 . . 3  |-  A  e. 
_V
4 indistopon 16754 . . 3  |-  ( A  e.  _V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
53, 4ax-mp 8 . 2  |-  { (/) ,  A }  e.  (TopOn `  A )
61eqcomi 2300 . . 3  |-  A  =  ( Base `  K
)
7 indistps2ALT.j . . . 4  |-  ( TopOpen `  K )  =  { (/)
,  A }
87eqcomi 2300 . . 3  |-  { (/) ,  A }  =  (
TopOpen `  K )
96, 8istps 16690 . 2  |-  ( K  e.  TopSp 
<->  { (/) ,  A }  e.  (TopOn `  A )
)
105, 9mpbir 200 1  |-  K  e. 
TopSp
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   {cpr 3654   ` cfv 5271   Basecbs 13164   TopOpenctopn 13342  TopOnctopon 16648   TopSpctps 16650
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-top 16652  df-topon 16655  df-topsp 16656
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