Theorem metustel 24688

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) → (𝐵 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑎))))

Distinct variable groups:   𝐵,𝑎   𝐷,𝑎   𝑋,𝑎

Allowed substitution hint:   𝐹(𝑎)

Proof of Theorem metustel

Step	Hyp	Ref	Expression
1		metust.1	. . 3 ⊢ 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
2	1	eleq2i 2855	. 2 ⊢ (𝐵 ∈ 𝐹 ↔ 𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))))
3		elex 3476	. . . 4 ⊢ (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) → 𝐵 ∈ V)
4	3	a1i 11	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) → 𝐵 ∈ V))
5		cnvexg 7922	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
6		imaexg 7911	. . . . 5 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑎)) ∈ V)
7		eleq1a 2858	. . . . 5 ⊢ ((◡𝐷 “ (0[,)𝑎)) ∈ V → (𝐵 = (◡𝐷 “ (0[,)𝑎)) → 𝐵 ∈ V))
8	5, 6, 7	3syl 19	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 = (◡𝐷 “ (0[,)𝑎)) → 𝐵 ∈ V))
9	8	rexlimdvw 3171	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑎 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑎)) → 𝐵 ∈ V))
10		eqid 2763	. . . . 5 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
11	10	elrnmpt 5950	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑎))))
12	11	a1i 11	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ V → (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑎)))))
13	4, 9, 12	pm5.21ndd 382	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑎))))
14	2, 13	bitrid 286	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑎))))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  Vcvv 3455   ↦ cmpt 5193  ^◡ccnv 5662  ran crn 5664   “ cima 5666  ‘cfv 6538  (class class class)co 7412  0cc0 11101  ℝ⁺crp 13017  [,)cico 13375  PsMetcpsmet 21487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is referenced by:  metustto  24691  metustid  24692  metustexhalf  24694  metustfbas  24695  cfilucfil  24697  metucn  24709