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Theorem 0totbnd 34938
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 6669 . . 3 (𝑋 = ∅ → (TotBnd‘𝑋) = (TotBnd‘∅))
21eleq2d 2903 . 2 (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (TotBnd‘∅)))
3 fveq2 6669 . . . 4 (𝑋 = ∅ → (Met‘𝑋) = (Met‘∅))
43eleq2d 2903 . . 3 (𝑋 = ∅ → (𝑀 ∈ (Met‘𝑋) ↔ 𝑀 ∈ (Met‘∅)))
5 0elpw 5253 . . . . . . 7 ∅ ∈ 𝒫 ∅
6 0fin 8740 . . . . . . 7 ∅ ∈ Fin
7 elin 4173 . . . . . . 7 (∅ ∈ (𝒫 ∅ ∩ Fin) ↔ (∅ ∈ 𝒫 ∅ ∧ ∅ ∈ Fin))
85, 6, 7mpbir2an 707 . . . . . 6 ∅ ∈ (𝒫 ∅ ∩ Fin)
9 0iun 4983 . . . . . 6 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅
10 iuneq1 4932 . . . . . . . 8 (𝑣 = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟))
1110eqeq1d 2828 . . . . . . 7 (𝑣 = ∅ → ( 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅ ↔ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅))
1211rspcev 3627 . . . . . 6 ((∅ ∈ (𝒫 ∅ ∩ Fin) ∧ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅) → ∃𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅)
138, 9, 12mp2an 688 . . . . 5 𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
1413rgenw 3155 . . . 4 𝑟 ∈ ℝ+𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
15 istotbnd3 34936 . . . 4 (𝑀 ∈ (TotBnd‘∅) ↔ (𝑀 ∈ (Met‘∅) ∧ ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅))
1614, 15mpbiran2 706 . . 3 (𝑀 ∈ (TotBnd‘∅) ↔ 𝑀 ∈ (Met‘∅))
174, 16syl6rbbr 291 . 2 (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘∅) ↔ 𝑀 ∈ (Met‘𝑋)))
182, 17bitrd 280 1 (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (Met‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  wcel 2107  wral 3143  wrex 3144  cin 3939  c0 4295  𝒫 cpw 4542   ciun 4917  cfv 6354  (class class class)co 7150  Fincfn 8503  +crp 12384  Metcmet 20466  ballcbl 20467  TotBndctotbnd 34931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8284  df-en 8504  df-dom 8505  df-fin 8507  df-totbnd 34933
This theorem is referenced by:  prdsbnd2  34960
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