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Theorem metustss 22684
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) ∧ 𝐴𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))
Distinct variable groups:   𝐷,𝑎   𝑋,𝑎   𝐴,𝑎   𝐹,𝑎

Proof of Theorem metustss
StepHypRef Expression
1 metust.1 . . . 4 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
2 cnvimass 5702 . . . . . . . . 9 (𝐷 “ (0[,)𝑎)) ⊆ dom 𝐷
3 psmetf 22439 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
42, 3fssdm 6272 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → (𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))
54ad2antrr 718 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) → (𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))
6 cnvexg 7347 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
7 imaexg 7338 . . . . . . . . 9 (𝐷 ∈ V → (𝐷 “ (0[,)𝑎)) ∈ V)
8 elpwg 4357 . . . . . . . . 9 ((𝐷 “ (0[,)𝑎)) ∈ V → ((𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)))
96, 7, 83syl 18 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → ((𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)))
109ad2antrr 718 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) → ((𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)))
115, 10mpbird 249 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) → (𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋))
1211ralrimiva 3147 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ∀𝑎 ∈ ℝ+ (𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋))
13 eqid 2799 . . . . . 6 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
1413rnmptss 6618 . . . . 5 (∀𝑎 ∈ ℝ+ (𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) → ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ⊆ 𝒫 (𝑋 × 𝑋))
1512, 14syl 17 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ⊆ 𝒫 (𝑋 × 𝑋))
161, 15syl5eqss 3845 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐹 ⊆ 𝒫 (𝑋 × 𝑋))
17 simpr 478 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴𝐹)
1816, 17sseldd 3799 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴 ∈ 𝒫 (𝑋 × 𝑋))
1918elpwid 4361 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  Vcvv 3385  wss 3769  𝒫 cpw 4349  cmpt 4922   × cxp 5310  ccnv 5311  ran crn 5313  cima 5315  cfv 6101  (class class class)co 6878  0cc0 10224  *cxr 10362  +crp 12074  [,)cico 12426  PsMetcpsmet 20052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097  df-xr 10367  df-psmet 20060
This theorem is referenced by:  metustrel  22685  metustsym  22688
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