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Mirrors > Home > MPE Home > Th. List > restsn | Structured version Visualization version GIF version |
Description: The only subspace topology induced by the topology {∅}. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
Ref | Expression |
---|---|
restsn | ⊢ (𝐴 ∈ 𝑉 → ({∅} ↾t 𝐴) = {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sn0top 22915 | . . . 4 ⊢ {∅} ∈ Top | |
2 | elrest 17409 | . . . 4 ⊢ (({∅} ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ ∃𝑦 ∈ {∅}𝑥 = (𝑦 ∩ 𝐴))) | |
3 | 1, 2 | mpan 689 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ ∃𝑦 ∈ {∅}𝑥 = (𝑦 ∩ 𝐴))) |
4 | 0ex 5307 | . . . . 5 ⊢ ∅ ∈ V | |
5 | ineq1 4205 | . . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 ∩ 𝐴) = (∅ ∩ 𝐴)) | |
6 | 0in 4394 | . . . . . . 7 ⊢ (∅ ∩ 𝐴) = ∅ | |
7 | 5, 6 | eqtrdi 2784 | . . . . . 6 ⊢ (𝑦 = ∅ → (𝑦 ∩ 𝐴) = ∅) |
8 | 7 | eqeq2d 2739 | . . . . 5 ⊢ (𝑦 = ∅ → (𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 = ∅)) |
9 | 4, 8 | rexsn 4687 | . . . 4 ⊢ (∃𝑦 ∈ {∅}𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 = ∅) |
10 | velsn 4645 | . . . 4 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
11 | 9, 10 | bitr4i 278 | . . 3 ⊢ (∃𝑦 ∈ {∅}𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 ∈ {∅}) |
12 | 3, 11 | bitrdi 287 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ 𝑥 ∈ {∅})) |
13 | 12 | eqrdv 2726 | 1 ⊢ (𝐴 ∈ 𝑉 → ({∅} ↾t 𝐴) = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ∃wrex 3067 ∩ cin 3946 ∅c0 4323 {csn 4629 (class class class)co 7420 ↾t crest 17402 Topctop 22808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-rest 17404 df-top 22809 df-topon 22826 |
This theorem is referenced by: (None) |
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