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Mirrors > Home > MPE Home > Th. List > istopg | Structured version Visualization version GIF version |
Description: Express the predicate
"𝐽 is a topology". See istop2g 22197 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 4572 | . . . . 5 ⊢ (𝑧 = 𝐽 → 𝒫 𝑧 = 𝒫 𝐽) | |
2 | eleq2 2826 | . . . . 5 ⊢ (𝑧 = 𝐽 → (∪ 𝑥 ∈ 𝑧 ↔ ∪ 𝑥 ∈ 𝐽)) | |
3 | 1, 2 | raleqbidv 3317 | . . . 4 ⊢ (𝑧 = 𝐽 → (∀𝑥 ∈ 𝒫 𝑧∪ 𝑥 ∈ 𝑧 ↔ ∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽)) |
4 | eleq2 2826 | . . . . . 6 ⊢ (𝑧 = 𝐽 → ((𝑥 ∩ 𝑦) ∈ 𝑧 ↔ (𝑥 ∩ 𝑦) ∈ 𝐽)) | |
5 | 4 | raleqbi1dv 3305 | . . . . 5 ⊢ (𝑧 = 𝐽 → (∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)) |
6 | 5 | raleqbi1dv 3305 | . . . 4 ⊢ (𝑧 = 𝐽 → (∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧 ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)) |
7 | 3, 6 | anbi12d 631 | . . 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 22195 | . . 3 ⊢ Top = {𝑧 ∣ (∀𝑥 ∈ 𝒫 𝑧∪ 𝑥 ∈ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)} | |
9 | 7, 8 | elab2g 3630 | . 2 ⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽 ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))) |
10 | df-ral 3063 | . . . 4 ⊢ (∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐽 → ∪ 𝑥 ∈ 𝐽)) | |
11 | elpw2g 5299 | . . . . . 6 ⊢ (𝐽 ∈ 𝐴 → (𝑥 ∈ 𝒫 𝐽 ↔ 𝑥 ⊆ 𝐽)) | |
12 | 11 | imbi1d 341 | . . . . 5 ⊢ (𝐽 ∈ 𝐴 → ((𝑥 ∈ 𝒫 𝐽 → ∪ 𝑥 ∈ 𝐽) ↔ (𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽))) |
13 | 12 | albidv 1923 | . . . 4 ⊢ (𝐽 ∈ 𝐴 → (∀𝑥(𝑥 ∈ 𝒫 𝐽 → ∪ 𝑥 ∈ 𝐽) ↔ ∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽))) |
14 | 10, 13 | bitrid 282 | . . 3 ⊢ (𝐽 ∈ 𝐴 → (∀𝑥 ∈ 𝒫 𝐽∪ 𝑥 ∈ 𝐽 ↔ ∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽))) |
15 | 14 | anbi1d 630 | . 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 278 | 1 ⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ∀wral 3062 ∩ cin 3907 ⊆ wss 3908 𝒫 cpw 4558 ∪ cuni 4863 Topctop 22194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-sep 5254 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3063 df-rab 3406 df-v 3445 df-in 3915 df-ss 3925 df-pw 4560 df-top 22195 |
This theorem is referenced by: istop2g 22197 uniopn 22198 inopn 22200 tgcl 22271 distop 22297 indistopon 22303 fctop 22306 cctop 22308 ppttop 22309 epttop 22311 mretopd 22395 toponmre 22396 neiptoptop 22434 kgentopon 22841 qtoptop2 23002 filconn 23186 utoptop 23538 neibastop1 34769 |
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