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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 21497 | . . . . 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 5240 | . . . . . . . 8 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝒫 𝐽 ↔ 𝐴 ⊆ 𝐽)) | |
5 | 4 | biimpar 480 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → 𝐴 ∈ 𝒫 𝐽) |
6 | sseq1 3992 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐽 ↔ 𝐴 ⊆ 𝐽)) | |
7 | unieq 4840 | . . . . . . . . . 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 3595 | . . . . . . 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 3935 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4832 Topctop 21495 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 |
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-rex 3144 df-rab 3147 df-v 3497 df-in 3943 df-ss 3952 df-pw 4541 df-uni 4833 df-top 21496 |
This theorem is referenced by: iunopn 21500 unopn 21505 0opn 21506 topopn 21508 tgtop 21575 ntropn 21651 toponmre 21695 neips 21715 txcmplem1 22243 unimopn 23100 metrest 23128 cnopn 23389 locfinreflem 31099 cvmscld 32515 mblfinlem3 34925 mblfinlem4 34926 ismblfin 34927 |
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