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Theorem 0totbnd 33926
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 6374 . . 3 (𝑋 = ∅ → (TotBnd‘𝑋) = (TotBnd‘∅))
21eleq2d 2829 . 2 (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (TotBnd‘∅)))
3 fveq2 6374 . . . 4 (𝑋 = ∅ → (Met‘𝑋) = (Met‘∅))
43eleq2d 2829 . . 3 (𝑋 = ∅ → (𝑀 ∈ (Met‘𝑋) ↔ 𝑀 ∈ (Met‘∅)))
5 0elpw 4991 . . . . . . 7 ∅ ∈ 𝒫 ∅
6 0fin 8394 . . . . . . 7 ∅ ∈ Fin
7 elin 3957 . . . . . . 7 (∅ ∈ (𝒫 ∅ ∩ Fin) ↔ (∅ ∈ 𝒫 ∅ ∧ ∅ ∈ Fin))
85, 6, 7mpbir2an 702 . . . . . 6 ∅ ∈ (𝒫 ∅ ∩ Fin)
9 0iun 4732 . . . . . 6 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅
10 iuneq1 4689 . . . . . . . 8 (𝑣 = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟))
1110eqeq1d 2766 . . . . . . 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 683 . . . . 5 𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
1413rgenw 3070 . . . 4 𝑟 ∈ ℝ+𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
15 istotbnd3 33924 . . . 4 (𝑀 ∈ (TotBnd‘∅) ↔ (𝑀 ∈ (Met‘∅) ∧ ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅))
1614, 15mpbiran2 701 . . 3 (𝑀 ∈ (TotBnd‘∅) ↔ 𝑀 ∈ (Met‘∅))
174, 16syl6rbbr 281 . 2 (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘∅) ↔ 𝑀 ∈ (Met‘𝑋)))
182, 17bitrd 270 1 (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (Met‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1652  wcel 2155  wral 3054  wrex 3055  cin 3730  c0 4078  𝒫 cpw 4314   ciun 4675  cfv 6067  (class class class)co 6841  Fincfn 8159  +crp 12027  Metcmet 20004  ballcbl 20005  TotBndctotbnd 33919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-int 4633  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-oadd 7767  df-er 7946  df-en 8160  df-dom 8161  df-fin 8163  df-totbnd 33921
This theorem is referenced by:  prdsbnd2  33948
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