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Theorem metustss 23903
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 6032 . . . . . . . . 9 (𝐷 “ (0[,)𝑎)) ⊆ dom 𝐷
3 psmetf 23655 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
42, 3fssdm 6686 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → (𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))
54ad2antrr 724 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) → (𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋))
6 cnvexg 7858 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
7 imaexg 7849 . . . . . . . . 9 (𝐷 ∈ V → (𝐷 “ (0[,)𝑎)) ∈ V)
8 elpwg 4562 . . . . . . . . 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 724 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) → ((𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋) ↔ (𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)))
115, 10mpbird 256 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) → (𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋))
1211ralrimiva 3142 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ∀𝑎 ∈ ℝ+ (𝐷 “ (0[,)𝑎)) ∈ 𝒫 (𝑋 × 𝑋))
13 eqid 2736 . . . . . 6 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
1413rnmptss 7067 . . . . 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 3991 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐹 ⊆ 𝒫 (𝑋 × 𝑋))
17 simpr 485 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴𝐹)
1816, 17sseldd 3944 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴 ∈ 𝒫 (𝑋 × 𝑋))
1918elpwid 4568 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3063  Vcvv 3444  wss 3909  𝒫 cpw 4559  cmpt 5187   × cxp 5630  ccnv 5631  ran crn 5633  cima 5635  cfv 6494  (class class class)co 7354  0cc0 11048  *cxr 11185  +crp 12912  [,)cico 13263  PsMetcpsmet 20776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7669  ax-cnex 11104  ax-resscn 11105
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-fv 6502  df-ov 7357  df-oprab 7358  df-mpo 7359  df-map 8764  df-xr 11190  df-psmet 20784
This theorem is referenced by:  metustrel  23904  metustsym  23907
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