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Theorem indistps2ALT 21768
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 21766 from the structural version indistps 21765. (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 ∈ TopSp

Proof of Theorem indistps2ALT
StepHypRef Expression
1 indistps2ALT.a . . . 4 (Base‘𝐾) = 𝐴
2 fvex 6690 . . . 4 (Base‘𝐾) ∈ V
31, 2eqeltrri 2831 . . 3 𝐴 ∈ V
4 indistopon 21755 . . 3 (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴))
53, 4ax-mp 5 . 2 {∅, 𝐴} ∈ (TopOn‘𝐴)
61eqcomi 2748 . . 3 𝐴 = (Base‘𝐾)
7 indistps2ALT.j . . . 4 (TopOpen‘𝐾) = {∅, 𝐴}
87eqcomi 2748 . . 3 {∅, 𝐴} = (TopOpen‘𝐾)
96, 8istps 21688 . 2 (𝐾 ∈ TopSp ↔ {∅, 𝐴} ∈ (TopOn‘𝐴))
105, 9mpbir 234 1 𝐾 ∈ TopSp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2114  Vcvv 3399  c0 4212  {cpr 4519  cfv 6340  Basecbs 16589  TopOpenctopn 16801  TopOnctopon 21664  TopSpctps 21686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-iota 6298  df-fun 6342  df-fv 6348  df-top 21648  df-topon 21665  df-topsp 21687
This theorem is referenced by: (None)
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