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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 4127 | . . . 4 ⊢ ∅ ∈ V | |
2 | restval 12639 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ ∅ ∈ V) → (𝐽 ↾t ∅) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅))) | |
3 | 1, 2 | mpan2 425 | . . 3 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅))) |
4 | in0 3457 | . . . . . . 7 ⊢ (𝑥 ∩ ∅) = ∅ | |
5 | 1 | elsn2 3625 | . . . . . . 7 ⊢ ((𝑥 ∩ ∅) ∈ {∅} ↔ (𝑥 ∩ ∅) = ∅) |
6 | 4, 5 | mpbir 146 | . . . . . 6 ⊢ (𝑥 ∩ ∅) ∈ {∅} |
7 | 6 | a1i 9 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ ∅) ∈ {∅}) |
8 | 7 | fmpttd 5667 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅)):𝐽⟶{∅}) |
9 | 8 | frnd 5371 | . . 3 ⊢ (𝐽 ∈ Top → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅)) ⊆ {∅}) |
10 | 3, 9 | eqsstrd 3191 | . 2 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) ⊆ {∅}) |
11 | resttop 13330 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ ∅ ∈ V) → (𝐽 ↾t ∅) ∈ Top) | |
12 | 1, 11 | mpan2 425 | . . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) ∈ Top) |
13 | 0opn 13164 | . . . 4 ⊢ ((𝐽 ↾t ∅) ∈ Top → ∅ ∈ (𝐽 ↾t ∅)) | |
14 | 12, 13 | syl 14 | . . 3 ⊢ (𝐽 ∈ Top → ∅ ∈ (𝐽 ↾t ∅)) |
15 | 14 | snssd 3736 | . 2 ⊢ (𝐽 ∈ Top → {∅} ⊆ (𝐽 ↾t ∅)) |
16 | 10, 15 | eqssd 3172 | 1 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) = {∅}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∩ cin 3128 ∅c0 3422 {csn 3591 ↦ cmpt 4061 ran crn 4624 (class class class)co 5869 ↾t crest 12633 Topctop 13155 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-rest 12635 df-topgen 12654 df-top 13156 df-bases 13201 |
This theorem is referenced by: (None) |
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