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Mirrors > Home > ILE Home > Th. List > inopn | GIF version |
Description: The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
Ref | Expression |
---|---|
inopn | ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istopg 11752 | . . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))) | |
2 | 1 | ibi 175 | . . . 4 ⊢ (𝐽 ∈ Top → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)) |
3 | 2 | simprd 113 | . . 3 ⊢ (𝐽 ∈ Top → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽) |
4 | ineq1 3195 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦)) | |
5 | 4 | eleq1d 2157 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑦) ∈ 𝐽 ↔ (𝐴 ∩ 𝑦) ∈ 𝐽)) |
6 | ineq2 3196 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐵)) | |
7 | 6 | eleq1d 2157 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∩ 𝑦) ∈ 𝐽 ↔ (𝐴 ∩ 𝐵) ∈ 𝐽)) |
8 | 5, 7 | rspc2v 2735 | . . 3 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽 → (𝐴 ∩ 𝐵) ∈ 𝐽)) |
9 | 3, 8 | syl5com 29 | . 2 ⊢ (𝐽 ∈ Top → ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽)) |
10 | 9 | 3impib 1142 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 925 ∀wal 1288 = wceq 1290 ∈ wcel 1439 ∀wral 2360 ∩ cin 2999 ⊆ wss 3000 ∪ cuni 3659 Topctop 11750 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-v 2622 df-in 3006 df-ss 3013 df-pw 3435 df-top 11751 |
This theorem is referenced by: tgclb 11819 topbas 11821 difopn 11862 uncld 11867 ntrin 11878 |
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