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Theorem indistps2ALT 19699
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 19697 from the structural version indistps 19696. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (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 5815 . . . 4  |-  ( Base `  K )  e.  _V
31, 2eqeltrri 2487 . . 3  |-  A  e. 
_V
4 indistopon 19686 . . 3  |-  ( A  e.  _V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
53, 4ax-mp 5 . 2  |-  { (/) ,  A }  e.  (TopOn `  A )
61eqcomi 2415 . . 3  |-  A  =  ( Base `  K
)
7 indistps2ALT.j . . . 4  |-  ( TopOpen `  K )  =  { (/)
,  A }
87eqcomi 2415 . . 3  |-  { (/) ,  A }  =  (
TopOpen `  K )
96, 8istps 19621 . 2  |-  ( K  e.  TopSp 
<->  { (/) ,  A }  e.  (TopOn `  A )
)
105, 9mpbir 209 1  |-  K  e. 
TopSp
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    e. wcel 1842   _Vcvv 3058   (/)c0 3737   {cpr 3973   ` cfv 5525   Basecbs 14733   TopOpenctopn 14928  TopOnctopon 19579   TopSpctps 19581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-iota 5489  df-fun 5527  df-fv 5533  df-top 19583  df-topon 19586  df-topsp 19587
This theorem is referenced by: (None)
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