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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 23021 | . . . 4 ⊢ {∅} ∈ Top | |
2 | elrest 17473 | . . . 4 ⊢ (({∅} ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ ∃𝑦 ∈ {∅}𝑥 = (𝑦 ∩ 𝐴))) | |
3 | 1, 2 | mpan 690 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ ∃𝑦 ∈ {∅}𝑥 = (𝑦 ∩ 𝐴))) |
4 | 0ex 5312 | . . . . 5 ⊢ ∅ ∈ V | |
5 | ineq1 4220 | . . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 ∩ 𝐴) = (∅ ∩ 𝐴)) | |
6 | 0in 4402 | . . . . . . 7 ⊢ (∅ ∩ 𝐴) = ∅ | |
7 | 5, 6 | eqtrdi 2790 | . . . . . 6 ⊢ (𝑦 = ∅ → (𝑦 ∩ 𝐴) = ∅) |
8 | 7 | eqeq2d 2745 | . . . . 5 ⊢ (𝑦 = ∅ → (𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 = ∅)) |
9 | 4, 8 | rexsn 4686 | . . . 4 ⊢ (∃𝑦 ∈ {∅}𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 = ∅) |
10 | velsn 4646 | . . . 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 2732 | 1 ⊢ (𝐴 ∈ 𝑉 → ({∅} ↾t 𝐴) = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 ∩ cin 3961 ∅c0 4338 {csn 4630 (class class class)co 7430 ↾t crest 17466 Topctop 22914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-rest 17468 df-top 22915 df-topon 22932 |
This theorem is referenced by: (None) |
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