Theorem metustel 24515

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 2828	. 2 ⊢ (𝐵 ∈ 𝐹 ↔ 𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))))
3		elex 3450	. . . 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 7875	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
6		imaexg 7864	. . . . 5 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑎)) ∈ V)
7		eleq1a 2831	. . . . 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 3143	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑎 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑎)) → 𝐵 ∈ V))
10		eqid 2736	. . . . 5 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
11	10	elrnmpt 5913	. . . 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 3061  Vcvv 3429   ↦ cmpt 5166  ^◡ccnv 5630  ran crn 5632   “ cima 5634  ‘cfv 6498  (class class class)co 7367  0cc0 11038  ℝ⁺crp 12942  [,)cico 13300  PsMetcpsmet 21336

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 2708  ax-sep 5231  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644

This theorem is referenced by:  metustto  24518  metustid  24519  metustexhalf  24521  metustfbas  24522  cfilucfil  24524  metucn  24536