MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metustss Structured version   Visualization version   GIF version

Theorem metustss 24580
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 6102 . . . . . . . . 9 (𝐷 “ (0[,)𝑎)) ⊆ dom 𝐷
3 psmetf 24332 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
42, 3fssdm 6756 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → (𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))
54ad2antrr 726 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) → (𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))
6 cnvexg 7947 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
7 imaexg 7936 . . . . . . . . 9 (𝐷 ∈ V → (𝐷 “ (0[,)𝑎)) ∈ V)
8 elpwg 4608 . . . . . . . . 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 726 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) → ((𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)))
115, 10mpbird 257 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) → (𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋))
1211ralrimiva 3144 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ∀𝑎 ∈ ℝ+ (𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋))
13 eqid 2735 . . . . . 6 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
1413rnmptss 7143 . . . . 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, 15eqsstrid 4044 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐹 ⊆ 𝒫 (𝑋 × 𝑋))
17 simpr 484 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴𝐹)
1816, 17sseldd 3996 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴 ∈ 𝒫 (𝑋 × 𝑋))
1918elpwid 4614 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963  𝒫 cpw 4605  cmpt 5231   × cxp 5687  ccnv 5688  ran crn 5690  cima 5692  cfv 6563  (class class class)co 7431  0cc0 11153  *cxr 11292  +crp 13032  [,)cico 13386  PsMetcpsmet 21366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-xr 11297  df-psmet 21374
This theorem is referenced by:  metustrel  24581  metustsym  24584
  Copyright terms: Public domain W3C validator