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Theorem 0totbnd 33225
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 6150 . . 3 (𝑋 = ∅ → (TotBnd‘𝑋) = (TotBnd‘∅))
21eleq2d 2684 . 2 (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (TotBnd‘∅)))
3 fveq2 6150 . . . 4 (𝑋 = ∅ → (Met‘𝑋) = (Met‘∅))
43eleq2d 2684 . . 3 (𝑋 = ∅ → (𝑀 ∈ (Met‘𝑋) ↔ 𝑀 ∈ (Met‘∅)))
5 0elpw 4796 . . . . . . 7 ∅ ∈ 𝒫 ∅
6 0fin 8135 . . . . . . 7 ∅ ∈ Fin
7 elin 3776 . . . . . . 7 (∅ ∈ (𝒫 ∅ ∩ Fin) ↔ (∅ ∈ 𝒫 ∅ ∧ ∅ ∈ Fin))
85, 6, 7mpbir2an 954 . . . . . 6 ∅ ∈ (𝒫 ∅ ∩ Fin)
9 0iun 4545 . . . . . 6 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅
10 iuneq1 4502 . . . . . . . 8 (𝑣 = ∅ → 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟))
1110eqeq1d 2623 . . . . . . 7 (𝑣 = ∅ → ( 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅ ↔ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅))
1211rspcev 3295 . . . . . 6 ((∅ ∈ (𝒫 ∅ ∩ Fin) ∧ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅) → ∃𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅)
138, 9, 12mp2an 707 . . . . 5 𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
1413rgenw 2919 . . . 4 𝑟 ∈ ℝ+𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
15 istotbnd3 33223 . . . 4 (𝑀 ∈ (TotBnd‘∅) ↔ (𝑀 ∈ (Met‘∅) ∧ ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 ∅ ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑟) = ∅))
1614, 15mpbiran2 953 . . 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 1480  wcel 1987  wral 2907  wrex 2908  cin 3555  c0 3893  𝒫 cpw 4132   ciun 4487  cfv 5849  (class class class)co 6607  Fincfn 7902  +crp 11779  Metcme 19654  ballcbl 19655  TotBndctotbnd 33218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-oadd 7512  df-er 7690  df-en 7903  df-dom 7904  df-fin 7906  df-totbnd 33220
This theorem is referenced by:  prdsbnd2  33247
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