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Mirrors > Home > MPE Home > Th. List > uniopn | Structured version Visualization version GIF version |
Description: The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.) |
Ref | Expression |
---|---|
uniopn | ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → ∪ 𝐴 ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istopg 21502 | . . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))) | |
2 | 1 | ibi 269 | . . . 4 ⊢ (𝐽 ∈ Top → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)) |
3 | 2 | simpld 497 | . . 3 ⊢ (𝐽 ∈ Top → ∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽)) |
4 | elpw2g 5246 | . . . . . . . 8 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝒫 𝐽 ↔ 𝐴 ⊆ 𝐽)) | |
5 | 4 | biimpar 480 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → 𝐴 ∈ 𝒫 𝐽) |
6 | sseq1 3991 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐽 ↔ 𝐴 ⊆ 𝐽)) | |
7 | unieq 4848 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
8 | 7 | eleq1d 2897 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ 𝐽 ↔ ∪ 𝐴 ∈ 𝐽)) |
9 | 6, 8 | imbi12d 347 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ↔ (𝐴 ⊆ 𝐽 → ∪ 𝐴 ∈ 𝐽))) |
10 | 9 | spcgv 3594 | . . . . . . 7 ⊢ (𝐴 ∈ 𝒫 𝐽 → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) → (𝐴 ⊆ 𝐽 → ∪ 𝐴 ∈ 𝐽))) |
11 | 5, 10 | syl 17 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) → (𝐴 ⊆ 𝐽 → ∪ 𝐴 ∈ 𝐽))) |
12 | 11 | com23 86 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → (𝐴 ⊆ 𝐽 → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) → ∪ 𝐴 ∈ 𝐽))) |
13 | 12 | ex 415 | . . . 4 ⊢ (𝐽 ∈ Top → (𝐴 ⊆ 𝐽 → (𝐴 ⊆ 𝐽 → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) → ∪ 𝐴 ∈ 𝐽)))) |
14 | 13 | pm2.43d 53 | . . 3 ⊢ (𝐽 ∈ Top → (𝐴 ⊆ 𝐽 → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) → ∪ 𝐴 ∈ 𝐽))) |
15 | 3, 14 | mpid 44 | . 2 ⊢ (𝐽 ∈ Top → (𝐴 ⊆ 𝐽 → ∪ 𝐴 ∈ 𝐽)) |
16 | 15 | imp 409 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → ∪ 𝐴 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1531 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4538 ∪ cuni 4837 Topctop 21500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-in 3942 df-ss 3951 df-pw 4540 df-uni 4838 df-top 21501 |
This theorem is referenced by: iunopn 21505 unopn 21510 0opn 21511 topopn 21513 tgtop 21580 ntropn 21656 toponmre 21700 neips 21720 txcmplem1 22248 unimopn 23105 metrest 23133 cnopn 23394 locfinreflem 31104 cvmscld 32520 mblfinlem3 34930 mblfinlem4 34931 ismblfin 34932 |
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