Theorem metustel 24524

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 2829	. 2 ⊢ (𝐵 ∈ 𝐹 ↔ 𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))))
3		elex 3451	. . . 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 7866	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
6		imaexg 7855	. . . . 5 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑎)) ∈ V)
7		eleq1a 2832	. . . . 5 ⊢ ((◡𝐷 “ (0[,)𝑎)) ∈ V → (𝐵 = (◡𝐷 “ (0[,)𝑎)) → 𝐵 ∈ V))
8	5, 6, 7	3syl 18	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 = (◡𝐷 “ (0[,)𝑎)) → 𝐵 ∈ V))
9	8	rexlimdvw 3144	. . 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)))
11	10	elrnmpt 5905	. . . 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 379	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑎))))
14	2, 13	bitrid 283	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑎))))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  Vcvv 3430   ↦ cmpt 5167  ^◡ccnv 5621  ran crn 5623   “ cima 5625  ‘cfv 6490  (class class class)co 7358  0cc0 11027  ℝ⁺crp 12931  [,)cico 13289  PsMetcpsmet 21326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-xp 5628  df-rel 5629  df-cnv 5630  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635

This theorem is referenced by:  metustto  24527  metustid  24528  metustexhalf  24530  metustfbas  24531  cfilucfil  24533  metucn  24545