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Theorem metustel 23929
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 2826 . 2 (𝐡 ∈ 𝐹 ↔ 𝐡 ∈ ran (π‘Ž ∈ ℝ+ ↦ (◑𝐷 β€œ (0[,)π‘Ž))))
3 elex 3465 . . . 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 7865 . . . . 5 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ ◑𝐷 ∈ V)
6 imaexg 7856 . . . . 5 (◑𝐷 ∈ V β†’ (◑𝐷 β€œ (0[,)π‘Ž)) ∈ V)
7 eleq1a 2829 . . . . 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 3154 . . 3 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (βˆƒπ‘Ž ∈ ℝ+ 𝐡 = (◑𝐷 β€œ (0[,)π‘Ž)) β†’ 𝐡 ∈ V))
10 eqid 2733 . . . . 5 (π‘Ž ∈ ℝ+ ↦ (◑𝐷 β€œ (0[,)π‘Ž))) = (π‘Ž ∈ ℝ+ ↦ (◑𝐷 β€œ (0[,)π‘Ž)))
1110elrnmpt 5915 . . . 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 381 . 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 1542   ∈ wcel 2107  βˆƒwrex 3070  Vcvv 3447   ↦ cmpt 5192  β—‘ccnv 5636  ran crn 5638   β€œ cima 5640  β€˜cfv 6500  (class class class)co 7361  0cc0 11059  β„+crp 12923  [,)cico 13275  PsMetcpsmet 20803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-xp 5643  df-rel 5644  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650
This theorem is referenced by:  metustto  23932  metustid  23933  metustexhalf  23935  metustfbas  23936  cfilucfil  23938  metucn  23950
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