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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 5175 | . . . 4 ⊢ ∅ ∈ V | |
2 | restval 16692 | . . . 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 4299 | . . . . . . 7 ⊢ (𝑥 ∩ ∅) = ∅ | |
5 | 1 | elsn2 4564 | . . . . . . 7 ⊢ ((𝑥 ∩ ∅) ∈ {∅} ↔ (𝑥 ∩ ∅) = ∅) |
6 | 4, 5 | mpbir 234 | . . . . . 6 ⊢ (𝑥 ∩ ∅) ∈ {∅} |
7 | 6 | a1i 11 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ ∅) ∈ {∅}) |
8 | 7 | fmpttd 6856 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅)):𝐽⟶{∅}) |
9 | 8 | frnd 6494 | . . 3 ⊢ (𝐽 ∈ Top → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅)) ⊆ {∅}) |
10 | 3, 9 | eqsstrd 3953 | . 2 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) ⊆ {∅}) |
11 | resttop 21765 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ ∅ ∈ V) → (𝐽 ↾t ∅) ∈ Top) | |
12 | 1, 11 | mpan2 690 | . . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) ∈ Top) |
13 | 0opn 21509 | . . . 4 ⊢ ((𝐽 ↾t ∅) ∈ Top → ∅ ∈ (𝐽 ↾t ∅)) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝐽 ∈ Top → ∅ ∈ (𝐽 ↾t ∅)) |
15 | 14 | snssd 4702 | . 2 ⊢ (𝐽 ∈ Top → {∅} ⊆ (𝐽 ↾t ∅)) |
16 | 10, 15 | eqssd 3932 | 1 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 ∅c0 4243 {csn 4525 ↦ cmpt 5110 ran crn 5520 (class class class)co 7135 ↾t crest 16686 Topctop 21498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-oadd 8089 df-er 8272 df-en 8493 df-fin 8496 df-fi 8859 df-rest 16688 df-topgen 16709 df-top 21499 df-bases 21551 |
This theorem is referenced by: fiuncmp 22009 xkouni 22204 icccmp 23430 cncfiooicc 42536 |
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