Theorem metustel 24292

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 2824	. 2 ⊢ (𝐵 ∈ 𝐹 ↔ 𝐵 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))))
3		elex 3492	. . . 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 7919	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
6		imaexg 7910	. . . . 5 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑎)) ∈ V)
7		eleq1a 2827	. . . . 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 3159	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑎 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑎)) → 𝐵 ∈ V))
10		eqid 2731	. . . . 5 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
11	10	elrnmpt 5955	. . . 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 205   = wceq 1540   ∈ wcel 2105  ∃wrex 3069  Vcvv 3473   ↦ cmpt 5231  ^◡ccnv 5675  ran crn 5677   “ cima 5679  ‘cfv 6543  (class class class)co 7412  0cc0 11116  ℝ⁺crp 12981  [,)cico 13333  PsMetcpsmet 21132

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689

This theorem is referenced by:  metustto  24295  metustid  24296  metustexhalf  24298  metustfbas  24299  cfilucfil  24301  metucn  24313