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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 12166 | . . . . 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 3270 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦)) | |
5 | 4 | eleq1d 2208 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑦) ∈ 𝐽 ↔ (𝐴 ∩ 𝑦) ∈ 𝐽)) |
6 | ineq2 3271 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐵)) | |
7 | 6 | eleq1d 2208 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∩ 𝑦) ∈ 𝐽 ↔ (𝐴 ∩ 𝐵) ∈ 𝐽)) |
8 | 5, 7 | rspc2v 2802 | . . 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 1179 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 ∀wal 1329 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ∩ cin 3070 ⊆ wss 3071 ∪ 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-3an 964 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: tgclb 12234 topbas 12236 difopn 12277 uncld 12282 ntrin 12293 innei 12332 restopnb 12350 cnptoprest 12408 txcnp 12440 txcnmpt 12442 mopnin 12656 |
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