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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 16694 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐽) → (𝐵 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) | |
2 | df-ss 3898 | . . . . 5 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐵) | |
3 | eleq1 2877 | . . . . 5 ⊢ ((𝐵 ∩ 𝐴) = 𝐵 → ((𝐵 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ↔ 𝐵 ∈ (𝐽 ↾t 𝐴))) | |
4 | 2, 3 | sylbi 220 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 → ((𝐵 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ↔ 𝐵 ∈ (𝐽 ↾t 𝐴))) |
5 | 1, 4 | syl5ibcom 248 | . . 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 458 | 1 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ∈ (𝐽 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ⊆ wss 3881 (class class class)co 7135 ↾t crest 16686 Topctop 21498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-rest 16688 |
This theorem is referenced by: (None) |
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