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Theorem metustel 23133
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 2902 . 2 (𝐵𝐹𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))))
3 elex 3491 . . . 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 7605 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
6 imaexg 7596 . . . . 5 (𝐷 ∈ V → (𝐷 “ (0[,)𝑎)) ∈ V)
7 eleq1a 2906 . . . . 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 3277 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎)) → 𝐵 ∈ V))
10 eqid 2820 . . . . 5 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
1110elrnmpt 5802 . . . 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 383 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎))))
142, 13syl5bb 285 1 (𝐷 ∈ (PsMet‘𝑋) → (𝐵𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1537  wcel 2114  wrex 3126  Vcvv 3473  cmpt 5120  ccnv 5528  ran crn 5530  cima 5532  cfv 6329  (class class class)co 7131  0cc0 10513  +crp 12366  [,)cico 12717  PsMetcpsmet 20502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304  ax-un 7437
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4813  df-br 5041  df-opab 5103  df-mpt 5121  df-xp 5535  df-rel 5536  df-cnv 5537  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542
This theorem is referenced by:  metustto  23136  metustid  23137  metustexhalf  23139  metustfbas  23140  cfilucfil  23142  metucn  23154
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