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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 23006 | . . . 4 ⊢ {∅} ∈ Top | |
| 2 | elrest 17472 | . . . 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 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 5 | ineq1 4213 | . . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 ∩ 𝐴) = (∅ ∩ 𝐴)) | |
| 6 | 0in 4397 | . . . . . . 7 ⊢ (∅ ∩ 𝐴) = ∅ | |
| 7 | 5, 6 | eqtrdi 2793 | . . . . . 6 ⊢ (𝑦 = ∅ → (𝑦 ∩ 𝐴) = ∅) |
| 8 | 7 | eqeq2d 2748 | . . . . 5 ⊢ (𝑦 = ∅ → (𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 = ∅)) |
| 9 | 4, 8 | rexsn 4682 | . . . 4 ⊢ (∃𝑦 ∈ {∅}𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 = ∅) |
| 10 | velsn 4642 | . . . 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 2735 | 1 ⊢ (𝐴 ∈ 𝑉 → ({∅} ↾t 𝐴) = {∅}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ∩ cin 3950 ∅c0 4333 {csn 4626 (class class class)co 7431 ↾t crest 17465 Topctop 22899 |
| 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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-rest 17467 df-top 22900 df-topon 22917 |
| This theorem is referenced by: (None) |
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