MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metustel Structured version   Visualization version   GIF version

Theorem metustel 24403
Description: Define a filter base 𝐹 generated by a metric 𝐷. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Hypothesis
Ref Expression
metust.1 𝐹 = ran (π‘Ž ∈ ℝ+ ↦ (◑𝐷 β€œ (0[,)π‘Ž)))
Assertion
Ref Expression
metustel (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝐡 ∈ 𝐹 ↔ βˆƒπ‘Ž ∈ ℝ+ 𝐡 = (◑𝐷 β€œ (0[,)π‘Ž))))
Distinct variable groups:   𝐡,π‘Ž   𝐷,π‘Ž   𝑋,π‘Ž
Allowed substitution hint:   𝐹(π‘Ž)

Proof of Theorem metustel
StepHypRef Expression
1 metust.1 . . 3 𝐹 = ran (π‘Ž ∈ ℝ+ ↦ (◑𝐷 β€œ (0[,)π‘Ž)))
21eleq2i 2817 . 2 (𝐡 ∈ 𝐹 ↔ 𝐡 ∈ ran (π‘Ž ∈ ℝ+ ↦ (◑𝐷 β€œ (0[,)π‘Ž))))
3 elex 3485 . . . 4 (𝐡 ∈ ran (π‘Ž ∈ ℝ+ ↦ (◑𝐷 β€œ (0[,)π‘Ž))) β†’ 𝐡 ∈ V)
43a1i 11 . . 3 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝐡 ∈ ran (π‘Ž ∈ ℝ+ ↦ (◑𝐷 β€œ (0[,)π‘Ž))) β†’ 𝐡 ∈ V))
5 cnvexg 7909 . . . . 5 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ ◑𝐷 ∈ V)
6 imaexg 7900 . . . . 5 (◑𝐷 ∈ V β†’ (◑𝐷 β€œ (0[,)π‘Ž)) ∈ V)
7 eleq1a 2820 . . . . 5 ((◑𝐷 β€œ (0[,)π‘Ž)) ∈ V β†’ (𝐡 = (◑𝐷 β€œ (0[,)π‘Ž)) β†’ 𝐡 ∈ V))
85, 6, 73syl 18 . . . 4 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝐡 = (◑𝐷 β€œ (0[,)π‘Ž)) β†’ 𝐡 ∈ V))
98rexlimdvw 3152 . . 3 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (βˆƒπ‘Ž ∈ ℝ+ 𝐡 = (◑𝐷 β€œ (0[,)π‘Ž)) β†’ 𝐡 ∈ V))
10 eqid 2724 . . . . 5 (π‘Ž ∈ ℝ+ ↦ (◑𝐷 β€œ (0[,)π‘Ž))) = (π‘Ž ∈ ℝ+ ↦ (◑𝐷 β€œ (0[,)π‘Ž)))
1110elrnmpt 5946 . . . 4 (𝐡 ∈ V β†’ (𝐡 ∈ ran (π‘Ž ∈ ℝ+ ↦ (◑𝐷 β€œ (0[,)π‘Ž))) ↔ βˆƒπ‘Ž ∈ ℝ+ 𝐡 = (◑𝐷 β€œ (0[,)π‘Ž))))
1211a1i 11 . . 3 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝐡 ∈ V β†’ (𝐡 ∈ ran (π‘Ž ∈ ℝ+ ↦ (◑𝐷 β€œ (0[,)π‘Ž))) ↔ βˆƒπ‘Ž ∈ ℝ+ 𝐡 = (◑𝐷 β€œ (0[,)π‘Ž)))))
134, 9, 12pm5.21ndd 379 . 2 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝐡 ∈ ran (π‘Ž ∈ ℝ+ ↦ (◑𝐷 β€œ (0[,)π‘Ž))) ↔ βˆƒπ‘Ž ∈ ℝ+ 𝐡 = (◑𝐷 β€œ (0[,)π‘Ž))))
142, 13bitrid 283 1 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝐡 ∈ 𝐹 ↔ βˆƒπ‘Ž ∈ ℝ+ 𝐡 = (◑𝐷 β€œ (0[,)π‘Ž))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3062  Vcvv 3466   ↦ cmpt 5222  β—‘ccnv 5666  ran crn 5668   β€œ cima 5670  β€˜cfv 6534  (class class class)co 7402  0cc0 11107  β„+crp 12975  [,)cico 13327  PsMetcpsmet 21218
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680
This theorem is referenced by:  metustto  24406  metustid  24407  metustexhalf  24409  metustfbas  24410  cfilucfil  24412  metucn  24424
  Copyright terms: Public domain W3C validator