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Mirrors > Home > ILE Home > Th. List > rest0 | GIF version |
Description: The subspace topology induced by the topology 𝐽 on the empty set. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 1-May-2015.) |
Ref | Expression |
---|---|
rest0 | ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) = {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4103 | . . . 4 ⊢ ∅ ∈ V | |
2 | restval 12498 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ ∅ ∈ V) → (𝐽 ↾t ∅) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅))) | |
3 | 1, 2 | mpan2 422 | . . 3 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅))) |
4 | in0 3438 | . . . . . . 7 ⊢ (𝑥 ∩ ∅) = ∅ | |
5 | 1 | elsn2 3604 | . . . . . . 7 ⊢ ((𝑥 ∩ ∅) ∈ {∅} ↔ (𝑥 ∩ ∅) = ∅) |
6 | 4, 5 | mpbir 145 | . . . . . 6 ⊢ (𝑥 ∩ ∅) ∈ {∅} |
7 | 6 | a1i 9 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ ∅) ∈ {∅}) |
8 | 7 | fmpttd 5634 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅)):𝐽⟶{∅}) |
9 | 8 | frnd 5341 | . . 3 ⊢ (𝐽 ∈ Top → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅)) ⊆ {∅}) |
10 | 3, 9 | eqsstrd 3173 | . 2 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) ⊆ {∅}) |
11 | resttop 12711 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ ∅ ∈ V) → (𝐽 ↾t ∅) ∈ Top) | |
12 | 1, 11 | mpan2 422 | . . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) ∈ Top) |
13 | 0opn 12545 | . . . 4 ⊢ ((𝐽 ↾t ∅) ∈ Top → ∅ ∈ (𝐽 ↾t ∅)) | |
14 | 12, 13 | syl 14 | . . 3 ⊢ (𝐽 ∈ Top → ∅ ∈ (𝐽 ↾t ∅)) |
15 | 14 | snssd 3712 | . 2 ⊢ (𝐽 ∈ Top → {∅} ⊆ (𝐽 ↾t ∅)) |
16 | 10, 15 | eqssd 3154 | 1 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) = {∅}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ∩ cin 3110 ∅c0 3404 {csn 3570 ↦ cmpt 4037 ran crn 4599 (class class class)co 5836 ↾t crest 12492 Topctop 12536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-rest 12494 df-topgen 12513 df-top 12537 df-bases 12582 |
This theorem is referenced by: (None) |
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