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| Mirrors > Home > MPE Home > Th. List > indistps2ALT | Structured version Visualization version GIF version | ||
| 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 23020 from the structural version indistps 23019. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| indistps2ALT.a | ⊢ (Base‘𝐾) = 𝐴 | 
| indistps2ALT.j | ⊢ (TopOpen‘𝐾) = {∅, 𝐴} | 
| Ref | Expression | 
|---|---|
| indistps2ALT | ⊢ 𝐾 ∈ TopSp | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | indistps2ALT.a | . . . 4 ⊢ (Base‘𝐾) = 𝐴 | |
| 2 | fvex 6918 | . . . 4 ⊢ (Base‘𝐾) ∈ V | |
| 3 | 1, 2 | eqeltrri 2837 | . . 3 ⊢ 𝐴 ∈ V | 
| 4 | indistopon 23009 | . . 3 ⊢ (𝐴 ∈ V → {∅, 𝐴} ∈ (TopOn‘𝐴)) | |
| 5 | 3, 4 | ax-mp 5 | . 2 ⊢ {∅, 𝐴} ∈ (TopOn‘𝐴) | 
| 6 | 1 | eqcomi 2745 | . . 3 ⊢ 𝐴 = (Base‘𝐾) | 
| 7 | indistps2ALT.j | . . . 4 ⊢ (TopOpen‘𝐾) = {∅, 𝐴} | |
| 8 | 7 | eqcomi 2745 | . . 3 ⊢ {∅, 𝐴} = (TopOpen‘𝐾) | 
| 9 | 6, 8 | istps 22941 | . 2 ⊢ (𝐾 ∈ TopSp ↔ {∅, 𝐴} ∈ (TopOn‘𝐴)) | 
| 10 | 5, 9 | mpbir 231 | 1 ⊢ 𝐾 ∈ TopSp | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∅c0 4332 {cpr 4627 ‘cfv 6560 Basecbs 17248 TopOpenctopn 17467 TopOnctopon 22917 TopSpctps 22939 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-top 22901 df-topon 22918 df-topsp 22940 | 
| This theorem is referenced by: (None) | 
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