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Mirrors > Home > MPE Home > Th. List > Mathboxes > toprestsubel | Structured version Visualization version GIF version |
Description: A subset is open in the topology it generates via restriction. (Contributed by Glauco Siliprandi, 21-Dec-2024.) |
Ref | Expression |
---|---|
toprestsubel.1 | ⊢ (𝜑 → 𝐽 ∈ Top) |
toprestsubel.2 | ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽) |
Ref | Expression |
---|---|
toprestsubel | ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toprestsubel.1 | . 2 ⊢ (𝜑 → 𝐽 ∈ Top) | |
2 | eqid 2736 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
3 | 2 | topopn 22902 | . . 3 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝜑 → ∪ 𝐽 ∈ 𝐽) |
5 | toprestsubel.2 | . 2 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽) | |
6 | 1, 4, 5 | restsubel 45131 | 1 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 ⊆ wss 3950 ∪ cuni 4905 (class class class)co 7429 ↾t crest 17461 Topctop 22889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pr 5430 ax-un 7751 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-ov 7432 df-oprab 7433 df-mpo 7434 df-rest 17463 df-top 22890 |
This theorem is referenced by: (None) |
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