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Theorem 0totbnd 32742
 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 6103 . . 3 (𝑋 = ∅ → (TotBnd‘𝑋) = (TotBnd‘∅))
21eleq2d 2673 . 2 (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (TotBnd‘∅)))
3 fveq2 6103 . . . 4 (𝑋 = ∅ → (Met‘𝑋) = (Met‘∅))
43eleq2d 2673 . . 3 (𝑋 = ∅ → (𝑀 ∈ (Met‘𝑋) ↔ 𝑀 ∈ (Met‘∅)))
5 0elpw 4760 . . . . . . 7 ∅ ∈ 𝒫 ∅
6 0fin 8073 . . . . . . 7 ∅ ∈ Fin
7 elin 3758 . . . . . . 7 (∅ ∈ (𝒫 ∅ ∩ Fin) ↔ (∅ ∈ 𝒫 ∅ ∧ ∅ ∈ Fin))
85, 6, 7mpbir2an 957 . . . . . 6 ∅ ∈ (𝒫 ∅ ∩ Fin)
9 0iun 4513 . . . . . 6 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅
10 iuneq1 4470 . . . . . . . 8 (𝑣 = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟))
1110eqeq1d 2612 . . . . . . 7 (𝑣 = ∅ → ( 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅ ↔ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅))
1211rspcev 3282 . . . . . 6 ((∅ ∈ (𝒫 ∅ ∩ Fin) ∧ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅) → ∃𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅)
138, 9, 12mp2an 704 . . . . 5 𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
1413rgenw 2908 . . . 4 𝑟 ∈ ℝ+𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
15 istotbnd3 32740 . . . 4 (𝑀 ∈ (TotBnd‘∅) ↔ (𝑀 ∈ (Met‘∅) ∧ ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅))
1614, 15mpbiran2 956 . . 3 (𝑀 ∈ (TotBnd‘∅) ↔ 𝑀 ∈ (Met‘∅))
174, 16syl6rbbr 278 . 2 (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘∅) ↔ 𝑀 ∈ (Met‘𝑋)))
182, 17bitrd 267 1 (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (Met‘𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539  ∅c0 3874  𝒫 cpw 4108  ∪ ciun 4455  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  ℝ+crp 11708  Metcme 19553  ballcbl 19554  TotBndctotbnd 32735 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-fin 7845  df-totbnd 32737 This theorem is referenced by:  prdsbnd2  32764
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