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Theorem 0totbnd 33904
Description: The metric (there is only one) on the empty set is totally bounded. (Contributed by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
0totbnd (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (Met‘𝑋)))

Proof of Theorem 0totbnd
Dummy variables 𝑣 𝑟 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6332 . . 3 (𝑋 = ∅ → (TotBnd‘𝑋) = (TotBnd‘∅))
21eleq2d 2836 . 2 (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (TotBnd‘∅)))
3 fveq2 6332 . . . 4 (𝑋 = ∅ → (Met‘𝑋) = (Met‘∅))
43eleq2d 2836 . . 3 (𝑋 = ∅ → (𝑀 ∈ (Met‘𝑋) ↔ 𝑀 ∈ (Met‘∅)))
5 0elpw 4965 . . . . . . 7 ∅ ∈ 𝒫 ∅
6 0fin 8344 . . . . . . 7 ∅ ∈ Fin
7 elin 3947 . . . . . . 7 (∅ ∈ (𝒫 ∅ ∩ Fin) ↔ (∅ ∈ 𝒫 ∅ ∧ ∅ ∈ Fin))
85, 6, 7mpbir2an 690 . . . . . 6 ∅ ∈ (𝒫 ∅ ∩ Fin)
9 0iun 4711 . . . . . 6 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅
10 iuneq1 4668 . . . . . . . 8 (𝑣 = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟))
1110eqeq1d 2773 . . . . . . 7 (𝑣 = ∅ → ( 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅ ↔ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅))
1211rspcev 3460 . . . . . 6 ((∅ ∈ (𝒫 ∅ ∩ Fin) ∧ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅) → ∃𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅)
138, 9, 12mp2an 672 . . . . 5 𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
1413rgenw 3073 . . . 4 𝑟 ∈ ℝ+𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
15 istotbnd3 33902 . . . 4 (𝑀 ∈ (TotBnd‘∅) ↔ (𝑀 ∈ (Met‘∅) ∧ ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅))
1614, 15mpbiran2 689 . . 3 (𝑀 ∈ (TotBnd‘∅) ↔ 𝑀 ∈ (Met‘∅))
174, 16syl6rbbr 279 . 2 (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘∅) ↔ 𝑀 ∈ (Met‘𝑋)))
182, 17bitrd 268 1 (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (Met‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  wral 3061  wrex 3062  cin 3722  c0 4063  𝒫 cpw 4297   ciun 4654  cfv 6031  (class class class)co 6793  Fincfn 8109  +crp 12035  Metcme 19947  ballcbl 19948  TotBndctotbnd 33897
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-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-3or 1072  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-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  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-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  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-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-fin 8113  df-totbnd 33899
This theorem is referenced by:  prdsbnd2  33926
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