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Theorem restsn 23178
Description: The only subspace topology induced by the topology {∅}. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
Assertion
Ref Expression
restsn (𝐴𝑉 → ({∅} ↾t 𝐴) = {∅})

Proof of Theorem restsn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sn0top 23006 . . . 4 {∅} ∈ Top
2 elrest 17472 . . . 4 (({∅} ∈ Top ∧ 𝐴𝑉) → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ ∃𝑦 ∈ {∅}𝑥 = (𝑦𝐴)))
31, 2mpan 690 . . 3 (𝐴𝑉 → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ ∃𝑦 ∈ {∅}𝑥 = (𝑦𝐴)))
4 0ex 5307 . . . . 5 ∅ ∈ V
5 ineq1 4213 . . . . . . 7 (𝑦 = ∅ → (𝑦𝐴) = (∅ ∩ 𝐴))
6 0in 4397 . . . . . . 7 (∅ ∩ 𝐴) = ∅
75, 6eqtrdi 2793 . . . . . 6 (𝑦 = ∅ → (𝑦𝐴) = ∅)
87eqeq2d 2748 . . . . 5 (𝑦 = ∅ → (𝑥 = (𝑦𝐴) ↔ 𝑥 = ∅))
94, 8rexsn 4682 . . . 4 (∃𝑦 ∈ {∅}𝑥 = (𝑦𝐴) ↔ 𝑥 = ∅)
10 velsn 4642 . . . 4 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
119, 10bitr4i 278 . . 3 (∃𝑦 ∈ {∅}𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ {∅})
123, 11bitrdi 287 . 2 (𝐴𝑉 → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ 𝑥 ∈ {∅}))
1312eqrdv 2735 1 (𝐴𝑉 → ({∅} ↾t 𝐴) = {∅})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  wrex 3070  cin 3950  c0 4333  {csn 4626  (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-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-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  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-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-rest 17467  df-top 22900  df-topon 22917
This theorem is referenced by: (None)
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