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Theorem restsn 21195
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 21024 . . . 4 {∅} ∈ Top
2 elrest 16296 . . . 4 (({∅} ∈ Top ∧ 𝐴𝑉) → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ ∃𝑦 ∈ {∅}𝑥 = (𝑦𝐴)))
31, 2mpan 670 . . 3 (𝐴𝑉 → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ ∃𝑦 ∈ {∅}𝑥 = (𝑦𝐴)))
4 0ex 4924 . . . . 5 ∅ ∈ V
5 ineq1 3958 . . . . . . 7 (𝑦 = ∅ → (𝑦𝐴) = (∅ ∩ 𝐴))
6 0in 4113 . . . . . . 7 (∅ ∩ 𝐴) = ∅
75, 6syl6eq 2821 . . . . . 6 (𝑦 = ∅ → (𝑦𝐴) = ∅)
87eqeq2d 2781 . . . . 5 (𝑦 = ∅ → (𝑥 = (𝑦𝐴) ↔ 𝑥 = ∅))
94, 8rexsn 4361 . . . 4 (∃𝑦 ∈ {∅}𝑥 = (𝑦𝐴) ↔ 𝑥 = ∅)
10 velsn 4332 . . . 4 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
119, 10bitr4i 267 . . 3 (∃𝑦 ∈ {∅}𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ {∅})
123, 11syl6bb 276 . 2 (𝐴𝑉 → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ 𝑥 ∈ {∅}))
1312eqrdv 2769 1 (𝐴𝑉 → ({∅} ↾t 𝐴) = {∅})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  wrex 3062  cin 3722  c0 4063  {csn 4316  (class class class)co 6793  t crest 16289  Topctop 20918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-rest 16291  df-top 20919  df-topon 20936
This theorem is referenced by: (None)
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