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Mirrors > Home > MPE Home > Th. List > rest0 | Structured version Visualization version 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 5306 | . . . 4 ⊢ ∅ ∈ V | |
2 | restval 17368 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ ∅ ∈ V) → (𝐽 ↾t ∅) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅))) | |
3 | 1, 2 | mpan2 690 | . . 3 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅))) |
4 | in0 4390 | . . . . . . 7 ⊢ (𝑥 ∩ ∅) = ∅ | |
5 | 1 | elsn2 4666 | . . . . . . 7 ⊢ ((𝑥 ∩ ∅) ∈ {∅} ↔ (𝑥 ∩ ∅) = ∅) |
6 | 4, 5 | mpbir 230 | . . . . . 6 ⊢ (𝑥 ∩ ∅) ∈ {∅} |
7 | 6 | a1i 11 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ ∅) ∈ {∅}) |
8 | 7 | fmpttd 7110 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅)):𝐽⟶{∅}) |
9 | 8 | frnd 6722 | . . 3 ⊢ (𝐽 ∈ Top → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅)) ⊆ {∅}) |
10 | 3, 9 | eqsstrd 4019 | . 2 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) ⊆ {∅}) |
11 | resttop 22646 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ ∅ ∈ V) → (𝐽 ↾t ∅) ∈ Top) | |
12 | 1, 11 | mpan2 690 | . . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) ∈ Top) |
13 | 0opn 22388 | . . . 4 ⊢ ((𝐽 ↾t ∅) ∈ Top → ∅ ∈ (𝐽 ↾t ∅)) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝐽 ∈ Top → ∅ ∈ (𝐽 ↾t ∅)) |
15 | 14 | snssd 4811 | . 2 ⊢ (𝐽 ∈ Top → {∅} ⊆ (𝐽 ↾t ∅)) |
16 | 10, 15 | eqssd 3998 | 1 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∩ cin 3946 ∅c0 4321 {csn 4627 ↦ cmpt 5230 ran crn 5676 (class class class)co 7404 ↾t crest 17362 Topctop 22377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-en 8936 df-fin 8939 df-fi 9402 df-rest 17364 df-topgen 17385 df-top 22378 df-bases 22431 |
This theorem is referenced by: fiuncmp 22890 xkouni 23085 icccmp 24323 cncfiooicc 44545 |
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