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Mirrors > Home > ILE Home > Th. List > restsn | 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 12040 | . . . 4 ⊢ {∅} ∈ Top | |
2 | elrest 11909 | . . . 4 ⊢ (({∅} ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ ∃𝑦 ∈ {∅}𝑥 = (𝑦 ∩ 𝐴))) | |
3 | 1, 2 | mpan 418 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ ∃𝑦 ∈ {∅}𝑥 = (𝑦 ∩ 𝐴))) |
4 | 0ex 3995 | . . . . 5 ⊢ ∅ ∈ V | |
5 | ineq1 3217 | . . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 ∩ 𝐴) = (∅ ∩ 𝐴)) | |
6 | 0in 3345 | . . . . . . 7 ⊢ (∅ ∩ 𝐴) = ∅ | |
7 | 5, 6 | syl6eq 2148 | . . . . . 6 ⊢ (𝑦 = ∅ → (𝑦 ∩ 𝐴) = ∅) |
8 | 7 | eqeq2d 2111 | . . . . 5 ⊢ (𝑦 = ∅ → (𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 = ∅)) |
9 | 4, 8 | rexsn 3515 | . . . 4 ⊢ (∃𝑦 ∈ {∅}𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 = ∅) |
10 | velsn 3491 | . . . 4 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) | |
11 | 9, 10 | bitr4i 186 | . . 3 ⊢ (∃𝑦 ∈ {∅}𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 ∈ {∅}) |
12 | 3, 11 | syl6bb 195 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ 𝑥 ∈ {∅})) |
13 | 12 | eqrdv 2098 | 1 ⊢ (𝐴 ∈ 𝑉 → ({∅} ↾t 𝐴) = {∅}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1299 ∈ wcel 1448 ∃wrex 2376 ∩ cin 3020 ∅c0 3310 {csn 3474 (class class class)co 5706 ↾t crest 11902 Topctop 11946 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-rest 11904 df-top 11947 df-topon 11960 |
This theorem is referenced by: (None) |
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