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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 5307 | . . . 4 ⊢ ∅ ∈ V | |
| 2 | restval 17471 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ ∅ ∈ V) → (𝐽 ↾t ∅) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅))) | |
| 3 | 1, 2 | mpan2 691 | . . 3 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅))) |
| 4 | in0 4395 | . . . . . . 7 ⊢ (𝑥 ∩ ∅) = ∅ | |
| 5 | 1 | elsn2 4665 | . . . . . . 7 ⊢ ((𝑥 ∩ ∅) ∈ {∅} ↔ (𝑥 ∩ ∅) = ∅) |
| 6 | 4, 5 | mpbir 231 | . . . . . 6 ⊢ (𝑥 ∩ ∅) ∈ {∅} |
| 7 | 6 | a1i 11 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ ∅) ∈ {∅}) |
| 8 | 7 | fmpttd 7135 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅)):𝐽⟶{∅}) |
| 9 | 8 | frnd 6744 | . . 3 ⊢ (𝐽 ∈ Top → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ ∅)) ⊆ {∅}) |
| 10 | 3, 9 | eqsstrd 4018 | . 2 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) ⊆ {∅}) |
| 11 | resttop 23168 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ ∅ ∈ V) → (𝐽 ↾t ∅) ∈ Top) | |
| 12 | 1, 11 | mpan2 691 | . . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) ∈ Top) |
| 13 | 0opn 22910 | . . . 4 ⊢ ((𝐽 ↾t ∅) ∈ Top → ∅ ∈ (𝐽 ↾t ∅)) | |
| 14 | 12, 13 | syl 17 | . . 3 ⊢ (𝐽 ∈ Top → ∅ ∈ (𝐽 ↾t ∅)) |
| 15 | 14 | snssd 4809 | . 2 ⊢ (𝐽 ∈ Top → {∅} ⊆ (𝐽 ↾t ∅)) |
| 16 | 10, 15 | eqssd 4001 | 1 ⊢ (𝐽 ∈ Top → (𝐽 ↾t ∅) = {∅}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ∅c0 4333 {csn 4626 ↦ cmpt 5225 ran crn 5686 (class class class)co 7431 ↾t crest 17465 Topctop 22899 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-en 8986 df-fin 8989 df-fi 9451 df-rest 17467 df-topgen 17488 df-top 22900 df-bases 22953 |
| This theorem is referenced by: fiuncmp 23412 xkouni 23607 icccmp 24847 cncfiooicc 45909 |
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