Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 20613 | . . . 4 ⊢ {∅} ∈ Top | |
2 | elrest 15911 | . . . 4 ⊢ (({∅} ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ ∃𝑦 ∈ {∅}𝑥 = (𝑦 ∩ 𝐴))) | |
3 | 1, 2 | mpan 702 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ ∃𝑦 ∈ {∅}𝑥 = (𝑦 ∩ 𝐴))) |
4 | 0ex 4718 | . . . . 5 ⊢ ∅ ∈ V | |
5 | ineq1 3769 | . . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 ∩ 𝐴) = (∅ ∩ 𝐴)) | |
6 | 0in 3921 | . . . . . . 7 ⊢ (∅ ∩ 𝐴) = ∅ | |
7 | 5, 6 | syl6eq 2660 | . . . . . 6 ⊢ (𝑦 = ∅ → (𝑦 ∩ 𝐴) = ∅) |
8 | 7 | eqeq2d 2620 | . . . . 5 ⊢ (𝑦 = ∅ → (𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 = ∅)) |
9 | 4, 8 | rexsn 4170 | . . . 4 ⊢ (∃𝑦 ∈ {∅}𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 = ∅) |
10 | velsn 4141 | . . . 4 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
11 | 9, 10 | bitr4i 266 | . . 3 ⊢ (∃𝑦 ∈ {∅}𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 ∈ {∅}) |
12 | 3, 11 | syl6bb 275 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ 𝑥 ∈ {∅})) |
13 | 12 | eqrdv 2608 | 1 ⊢ (𝐴 ∈ 𝑉 → ({∅} ↾t 𝐴) = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ∩ cin 3539 ∅c0 3874 {csn 4125 (class class class)co 6549 ↾t crest 15904 Topctop 20517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-rest 15906 df-top 20521 df-topon 20523 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |