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Theorem metustel 22103
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 2676 . 2 (𝐵𝐹𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))))
3 elex 3181 . . . 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 6979 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
6 imaexg 6969 . . . . 5 (𝐷 ∈ V → (𝐷 “ (0[,)𝑎)) ∈ V)
7 eleq1a 2679 . . . . 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 3012 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎)) → 𝐵 ∈ V))
10 eqid 2606 . . . . 5 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
1110elrnmpt 5277 . . . 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 367 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎))))
142, 13syl5bb 270 1 (𝐷 ∈ (PsMet‘𝑋) → (𝐵𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wcel 1976  wrex 2893  Vcvv 3169  cmpt 4634  ccnv 5024  ran crn 5026  cima 5028  cfv 5787  (class class class)co 6524  0cc0 9789  +crp 11661  [,)cico 12001  PsMetcpsmet 19494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-mpt 4636  df-xp 5031  df-rel 5032  df-cnv 5033  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038
This theorem is referenced by:  metustto  22106  metustid  22107  metustexhalf  22109  metustfbas  22110  cfilucfil  22112  metucn  22124
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