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Theorem metustel 24563
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 2833 . 2 (𝐵𝐹𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))))
3 elex 3501 . . . 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 7946 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
6 imaexg 7935 . . . . 5 (𝐷 ∈ V → (𝐷 “ (0[,)𝑎)) ∈ V)
7 eleq1a 2836 . . . . 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 3160 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎)) → 𝐵 ∈ V))
10 eqid 2737 . . . . 5 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
1110elrnmpt 5969 . . . 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 379 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎))))
142, 13bitrid 283 1 (𝐷 ∈ (PsMet‘𝑋) → (𝐵𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  cmpt 5225  ccnv 5684  ran crn 5686  cima 5688  cfv 6561  (class class class)co 7431  0cc0 11155  +crp 13034  [,)cico 13389  PsMetcpsmet 21348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by:  metustto  24566  metustid  24567  metustexhalf  24569  metustfbas  24570  cfilucfil  24572  metucn  24584
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