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Theorem metustss 22266
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 5444 . . . . . . . . 9 (𝐷 “ (0[,)𝑎)) ⊆ dom 𝐷
3 psmetf 22021 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
4 fdm 6008 . . . . . . . . . 10 (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋))
53, 4syl 17 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
62, 5syl5sseq 3632 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → (𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))
76ad2antrr 761 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) → (𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))
8 cnvexg 7059 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
9 imaexg 7050 . . . . . . . . 9 (𝐷 ∈ V → (𝐷 “ (0[,)𝑎)) ∈ V)
10 elpwg 4138 . . . . . . . . 9 ((𝐷 “ (0[,)𝑎)) ∈ V → ((𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)))
118, 9, 103syl 18 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → ((𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)))
1211ad2antrr 761 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) → ((𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)))
137, 12mpbird 247 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) → (𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋))
1413ralrimiva 2960 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ∀𝑎 ∈ ℝ+ (𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋))
15 eqid 2621 . . . . . 6 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
1615rnmptss 6347 . . . . 5 (∀𝑎 ∈ ℝ+ (𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) → ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ⊆ 𝒫 (𝑋 × 𝑋))
1714, 16syl 17 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ⊆ 𝒫 (𝑋 × 𝑋))
181, 17syl5eqss 3628 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐹 ⊆ 𝒫 (𝑋 × 𝑋))
19 simpr 477 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴𝐹)
2018, 19sseldd 3584 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴 ∈ 𝒫 (𝑋 × 𝑋))
2120elpwid 4141 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  wss 3555  𝒫 cpw 4130  cmpt 4673   × cxp 5072  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077  wf 5843  cfv 5847  (class class class)co 6604  0cc0 9880  *cxr 10017  +crp 11776  [,)cico 12119  PsMetcpsmet 19649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-xr 10022  df-psmet 19657
This theorem is referenced by:  metustrel  22267  metustsym  22270
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