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Mirrors > Home > MPE Home > Th. List > Mathboxes > subspopn | Structured version Visualization version GIF version |
Description: An open set is open in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
Ref | Expression |
---|---|
subspopn | ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ∈ (𝐽 ↾t 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrestr 17380 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐽) → (𝐵 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) | |
2 | df-ss 3960 | . . . . 5 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐵) | |
3 | eleq1 2815 | . . . . 5 ⊢ ((𝐵 ∩ 𝐴) = 𝐵 → ((𝐵 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ↔ 𝐵 ∈ (𝐽 ↾t 𝐴))) | |
4 | 2, 3 | sylbi 216 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 → ((𝐵 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ↔ 𝐵 ∈ (𝐽 ↾t 𝐴))) |
5 | 1, 4 | syl5ibcom 244 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐽) → (𝐵 ⊆ 𝐴 → 𝐵 ∈ (𝐽 ↾t 𝐴))) |
6 | 5 | 3expa 1115 | . 2 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ 𝐵 ∈ 𝐽) → (𝐵 ⊆ 𝐴 → 𝐵 ∈ (𝐽 ↾t 𝐴))) |
7 | 6 | impr 454 | 1 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ∈ (𝐽 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∩ cin 3942 ⊆ wss 3943 (class class class)co 7404 ↾t crest 17372 Topctop 22745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-rest 17374 |
This theorem is referenced by: (None) |
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