Theorem metustss 24564

Description: Range of the elements of the filter base generated by the metric 𝐷. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)

Hypothesis

Ref	Expression
metust.1	⊢ 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))

Assertion

Ref	Expression
metustss	⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))

Distinct variable groups:   𝐷,𝑎   𝑋,𝑎   𝐴,𝑎   𝐹,𝑎

Proof of Theorem metustss

Step	Hyp	Ref	Expression
1		metust.1	. . . 4 ⊢ 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
2		cnvimass 6100	. . . . . . . . 9 ⊢ (◡𝐷 “ (0[,)𝑎)) ⊆ dom 𝐷
3		psmetf 24316	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
4	2, 3	fssdm 6755	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))
5	4	ad2antrr 726	. . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) → (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))
6		cnvexg 7946	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
7		imaexg 7935	. . . . . . . . 9 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑎)) ∈ V)
8		elpwg 4603	. . . . . . . . 9 ⊢ ((◡𝐷 “ (0[,)𝑎)) ∈ V → ((◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)))
9	6, 7, 8	3syl 18	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ((◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)))
10	9	ad2antrr 726	. . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) → ((◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)))
11	5, 10	mpbird 257	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) → (◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋))
12	11	ralrimiva 3146	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ∀𝑎 ∈ ℝ+ (◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋))
13		eqid 2737	. . . . . 6 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
14	13	rnmptss 7143	. . . . 5 ⊢ (∀𝑎 ∈ ℝ+ (◡𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) → ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ⊆ 𝒫 (𝑋 × 𝑋))
15	12, 14	syl 17	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ⊆ 𝒫 (𝑋 × 𝑋))
16	1, 15	eqsstrid 4022	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐹 ⊆ 𝒫 (𝑋 × 𝑋))
17		simpr 484	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ∈ 𝐹)
18	16, 17	sseldd 3984	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ∈ 𝒫 (𝑋 × 𝑋))
19	18	elpwid 4609	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  Vcvv 3480   ⊆ wss 3951  𝒫 cpw 4600   ↦ cmpt 5225   × cxp 5683  ^◡ccnv 5684  ran crn 5686   “ cima 5688  ‘cfv 6561  (class class class)co 7431  0cc0 11155  ℝ^*cxr 11294  ℝ⁺crp 13034  [,)cico 13389  PsMetcpsmet 21348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-xr 11299  df-psmet 21356

This theorem is referenced by:  metustrel  24565  metustsym  24568