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Theorem metustid 22169
Description: The identity diagonal is included in all elements of the filter base generated by the metric 𝐷. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Hypothesis
Ref Expression
metust.1 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
Assertion
Ref Expression
metustid ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ( I ↾ 𝑋) ⊆ 𝐴)
Distinct variable groups:   𝐷,𝑎   𝑋,𝑎   𝐴,𝑎   𝐹,𝑎

Proof of Theorem metustid
Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 5346 . . 3 Rel ( I ↾ 𝑋)
21a1i 11 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → Rel ( I ↾ 𝑋))
3 vex 3176 . . . . . . . . . . . . . . 15 𝑞 ∈ V
43brres 5323 . . . . . . . . . . . . . 14 (𝑝( I ↾ 𝑋)𝑞 ↔ (𝑝 I 𝑞𝑝𝑋))
5 df-br 4584 . . . . . . . . . . . . . 14 (𝑝( I ↾ 𝑋)𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋))
63ideq 5196 . . . . . . . . . . . . . . 15 (𝑝 I 𝑞𝑝 = 𝑞)
76anbi1i 727 . . . . . . . . . . . . . 14 ((𝑝 I 𝑞𝑝𝑋) ↔ (𝑝 = 𝑞𝑝𝑋))
84, 5, 73bitr3i 289 . . . . . . . . . . . . 13 (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) ↔ (𝑝 = 𝑞𝑝𝑋))
98biimpi 205 . . . . . . . . . . . 12 (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) → (𝑝 = 𝑞𝑝𝑋))
109ad2antlr 759 . . . . . . . . . . 11 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝑝 = 𝑞𝑝𝑋))
1110simpld 474 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑝 = 𝑞)
12 df-ov 6552 . . . . . . . . . . 11 (𝑝𝐷𝑝) = (𝐷‘⟨𝑝, 𝑝⟩)
13 opeq2 4341 . . . . . . . . . . . 12 (𝑝 = 𝑞 → ⟨𝑝, 𝑝⟩ = ⟨𝑝, 𝑞⟩)
1413fveq2d 6107 . . . . . . . . . . 11 (𝑝 = 𝑞 → (𝐷‘⟨𝑝, 𝑝⟩) = (𝐷‘⟨𝑝, 𝑞⟩))
1512, 14syl5eq 2656 . . . . . . . . . 10 (𝑝 = 𝑞 → (𝑝𝐷𝑝) = (𝐷‘⟨𝑝, 𝑞⟩))
1611, 15syl 17 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝑝𝐷𝑝) = (𝐷‘⟨𝑝, 𝑞⟩))
17 simplll 794 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝐷 ∈ (PsMet‘𝑋))
1810simprd 478 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑝𝑋)
19 psmet0 21923 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑝𝑋) → (𝑝𝐷𝑝) = 0)
2017, 18, 19syl2anc 691 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝑝𝐷𝑝) = 0)
2116, 20eqtr3d 2646 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝐷‘⟨𝑝, 𝑞⟩) = 0)
22 0xr 9965 . . . . . . . . . . 11 0 ∈ ℝ*
2322a1i 11 . . . . . . . . . 10 (𝑎 ∈ ℝ+ → 0 ∈ ℝ*)
24 rpxr 11716 . . . . . . . . . 10 (𝑎 ∈ ℝ+𝑎 ∈ ℝ*)
25 rpgt0 11720 . . . . . . . . . 10 (𝑎 ∈ ℝ+ → 0 < 𝑎)
26 lbico1 12099 . . . . . . . . . 10 ((0 ∈ ℝ*𝑎 ∈ ℝ* ∧ 0 < 𝑎) → 0 ∈ (0[,)𝑎))
2723, 24, 25, 26syl3anc 1318 . . . . . . . . 9 (𝑎 ∈ ℝ+ → 0 ∈ (0[,)𝑎))
2827adantl 481 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 0 ∈ (0[,)𝑎))
2921, 28eqeltrd 2688 . . . . . . 7 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎))
30 psmetf 21921 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
31 ffun 5961 . . . . . . . . . 10 (𝐷:(𝑋 × 𝑋)⟶ℝ* → Fun 𝐷)
3230, 31syl 17 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → Fun 𝐷)
3332ad3antrrr 762 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → Fun 𝐷)
3411, 18eqeltrrd 2689 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑞𝑋)
35 opelxpi 5072 . . . . . . . . . 10 ((𝑝𝑋𝑞𝑋) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
3618, 34, 35syl2anc 691 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
37 fdm 5964 . . . . . . . . . . 11 (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋))
3830, 37syl 17 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
3938ad3antrrr 762 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → dom 𝐷 = (𝑋 × 𝑋))
4036, 39eleqtrrd 2691 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ dom 𝐷)
41 fvimacnv 6240 . . . . . . . 8 ((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎))))
4233, 40, 41syl2anc 691 . . . . . . 7 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎))))
4329, 42mpbid 221 . . . . . 6 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎)))
4443adantr 480 . . . . 5 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎)))
45 simpr 476 . . . . 5 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → 𝐴 = (𝐷 “ (0[,)𝑎)))
4644, 45eleqtrrd 2691 . . . 4 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ 𝐴)
47 simplr 788 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → 𝐴𝐹)
48 metust.1 . . . . . . 7 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
4948metustel 22165 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → (𝐴𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎))))
5049ad2antrr 758 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → (𝐴𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎))))
5147, 50mpbid 221 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)))
5246, 51r19.29a 3060 . . 3 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → ⟨𝑝, 𝑞⟩ ∈ 𝐴)
5352ex 449 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) → ⟨𝑝, 𝑞⟩ ∈ 𝐴))
542, 53relssdv 5135 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ( I ↾ 𝑋) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wrex 2897  wss 3540  cop 4131   class class class wbr 4583  cmpt 4643   I cid 4948   × cxp 5036  ccnv 5037  dom cdm 5038  ran crn 5039  cres 5040  cima 5041  Rel wrel 5043  Fun wfun 5798  wf 5800  cfv 5804  (class class class)co 6549  0cc0 9815  *cxr 9952   < clt 9953  +crp 11708  [,)cico 12048  PsMetcpsmet 19551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-rp 11709  df-ico 12052  df-psmet 19559
This theorem is referenced by:  metustfbas  22172  metust  22173
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