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Mirrors > Home > MPE Home > Th. List > distopon | Structured version Visualization version GIF version |
Description: The discrete topology on a set 𝐴, with base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
distopon | ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | distop 20890 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top) | |
2 | unipw 4991 | . . . 4 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
3 | 2 | eqcomi 2701 | . . 3 ⊢ 𝐴 = ∪ 𝒫 𝐴 |
4 | 3 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 = ∪ 𝒫 𝐴) |
5 | istopon 20808 | . 2 ⊢ (𝒫 𝐴 ∈ (TopOn‘𝐴) ↔ (𝒫 𝐴 ∈ Top ∧ 𝐴 = ∪ 𝒫 𝐴)) | |
6 | 1, 4, 5 | sylanbrc 701 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1564 ∈ wcel 2071 𝒫 cpw 4234 ∪ cuni 4512 ‘cfv 5969 Topctop 20789 TopOnctopon 20806 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-8 2073 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 ax-sep 4857 ax-nul 4865 ax-pow 4916 ax-pr 4979 ax-un 7034 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1567 df-ex 1786 df-nf 1791 df-sb 1979 df-eu 2543 df-mo 2544 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-ne 2865 df-ral 2987 df-rex 2988 df-rab 2991 df-v 3274 df-sbc 3510 df-dif 3651 df-un 3653 df-in 3655 df-ss 3662 df-nul 3992 df-if 4163 df-pw 4236 df-sn 4254 df-pr 4256 df-op 4260 df-uni 4513 df-br 4729 df-opab 4789 df-mpt 4806 df-id 5096 df-xp 5192 df-rel 5193 df-cnv 5194 df-co 5195 df-dm 5196 df-iota 5932 df-fun 5971 df-fv 5977 df-top 20790 df-topon 20807 |
This theorem is referenced by: sn0topon 20893 toponmre 20988 cndis 21186 txdis1cn 21529 xkofvcn 21578 distgp 21993 symgtgp 21995 cnfdmsn 40483 |
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