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Theorem metustel 24579
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 3499 . . . 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 7947 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
6 imaexg 7936 . . . . 5 (𝐷 ∈ V → (𝐷 “ (0[,)𝑎)) ∈ V)
7 eleq1a 2834 . . . . 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 3158 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎)) → 𝐵 ∈ V))
10 eqid 2735 . . . . 5 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
1110elrnmpt 5972 . . . 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 1537  wcel 2106  wrex 3068  Vcvv 3478  cmpt 5231  ccnv 5688  ran crn 5690  cima 5692  cfv 6563  (class class class)co 7431  0cc0 11153  +crp 13032  [,)cico 13386  PsMetcpsmet 21366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702
This theorem is referenced by:  metustto  24582  metustid  24583  metustexhalf  24585  metustfbas  24586  cfilucfil  24588  metucn  24600
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