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Theorem metustel 23479
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 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
Assertion
Ref Expression
metustel (𝐷 ∈ (PsMet‘𝑋) → (𝐵𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎))))
Distinct variable groups:   𝐵,𝑎   𝐷,𝑎   𝑋,𝑎
Allowed substitution hint:   𝐹(𝑎)

Proof of Theorem metustel
StepHypRef Expression
1 metust.1 . . 3 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
21eleq2i 2831 . 2 (𝐵𝐹𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))))
3 elex 3441 . . . 4 (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) → 𝐵 ∈ V)
43a1i 11 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) → 𝐵 ∈ V))
5 cnvexg 7723 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
6 imaexg 7714 . . . . 5 (𝐷 ∈ V → (𝐷 “ (0[,)𝑎)) ∈ V)
7 eleq1a 2835 . . . . 5 ((𝐷 “ (0[,)𝑎)) ∈ V → (𝐵 = (𝐷 “ (0[,)𝑎)) → 𝐵 ∈ V))
85, 6, 73syl 18 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (𝐵 = (𝐷 “ (0[,)𝑎)) → 𝐵 ∈ V))
98rexlimdvw 3219 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎)) → 𝐵 ∈ V))
10 eqid 2739 . . . . 5 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
1110elrnmpt 5842 . . . 4 (𝐵 ∈ V → (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎))))
1211a1i 11 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ V → (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎)))))
134, 9, 12pm5.21ndd 384 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎))))
142, 13syl5bb 286 1 (𝐷 ∈ (PsMet‘𝑋) → (𝐵𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wcel 2112  wrex 3065  Vcvv 3423  cmpt 5151  ccnv 5567  ran crn 5569  cima 5571  cfv 6400  (class class class)co 7234  0cc0 10758  +crp 12615  [,)cico 12966  PsMetcpsmet 20379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pow 5274  ax-pr 5338  ax-un 7544
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4456  df-pw 4531  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-br 5070  df-opab 5132  df-mpt 5152  df-xp 5574  df-rel 5575  df-cnv 5576  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581
This theorem is referenced by:  metustto  23482  metustid  23483  metustexhalf  23485  metustfbas  23486  cfilucfil  23488  metucn  23500
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