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Mirrors > Home > ILE Home > Th. List > istopg | GIF version |
Description: Express the predicate
"𝐽 is a topology". See istopfin 12167 for
another characterization using nonempty finite intersections instead of
binary intersections.
Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use 𝑇 to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
istopg | ⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweq 3513 | . . . . 5 ⊢ (𝑧 = 𝐽 → 𝒫 𝑧 = 𝒫 𝐽) | |
2 | eleq2 2203 | . . . . 5 ⊢ (𝑧 = 𝐽 → (∪ 𝑥 ∈ 𝑧 ↔ ∪ 𝑥 ∈ 𝐽)) | |
3 | 1, 2 | raleqbidv 2638 | . . . 4 ⊢ (𝑧 = 𝐽 → (∀𝑥 ∈ 𝒫 𝑧∪ 𝑥 ∈ 𝑧 ↔ ∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽)) |
4 | eleq2 2203 | . . . . . 6 ⊢ (𝑧 = 𝐽 → ((𝑥 ∩ 𝑦) ∈ 𝑧 ↔ (𝑥 ∩ 𝑦) ∈ 𝐽)) | |
5 | 4 | raleqbi1dv 2634 | . . . . 5 ⊢ (𝑧 = 𝐽 → (∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)) |
6 | 5 | raleqbi1dv 2634 | . . . 4 ⊢ (𝑧 = 𝐽 → (∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧 ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)) |
7 | 3, 6 | anbi12d 464 | . . 3 ⊢ (𝑧 = 𝐽 → ((∀𝑥 ∈ 𝒫 𝑧∪ 𝑥 ∈ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧) ↔ (∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽 ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))) |
8 | df-top 12165 | . . 3 ⊢ Top = {𝑧 ∣ (∀𝑥 ∈ 𝒫 𝑧∪ 𝑥 ∈ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)} | |
9 | 7, 8 | elab2g 2831 | . 2 ⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽 ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))) |
10 | df-ral 2421 | . . . 4 ⊢ (∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐽 → ∪ 𝑥 ∈ 𝐽)) | |
11 | elpw2g 4081 | . . . . . 6 ⊢ (𝐽 ∈ 𝐴 → (𝑥 ∈ 𝒫 𝐽 ↔ 𝑥 ⊆ 𝐽)) | |
12 | 11 | imbi1d 230 | . . . . 5 ⊢ (𝐽 ∈ 𝐴 → ((𝑥 ∈ 𝒫 𝐽 → ∪ 𝑥 ∈ 𝐽) ↔ (𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽))) |
13 | 12 | albidv 1796 | . . . 4 ⊢ (𝐽 ∈ 𝐴 → (∀𝑥(𝑥 ∈ 𝒫 𝐽 → ∪ 𝑥 ∈ 𝐽) ↔ ∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽))) |
14 | 10, 13 | syl5bb 191 | . . 3 ⊢ (𝐽 ∈ 𝐴 → (∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽 ↔ ∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽))) |
15 | 14 | anbi1d 460 | . 2 ⊢ (𝐽 ∈ 𝐴 → ((∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽 ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽) ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))) |
16 | 9, 15 | bitrd 187 | 1 ⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1329 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ∩ cin 3070 ⊆ wss 3071 𝒫 cpw 3510 ∪ cuni 3736 Topctop 12164 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-in 3077 df-ss 3084 df-pw 3512 df-top 12165 |
This theorem is referenced by: istopfin 12167 uniopn 12168 inopn 12170 tgcl 12233 distop 12254 epttop 12259 |
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