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Theorem metustid 24583
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.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
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 6026 . . 3 Rel ( I ↾ 𝑋)
21a1i 11 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → Rel ( I ↾ 𝑋))
3 vex 3482 . . . . . . . . . . . . . . 15 𝑞 ∈ V
43brresi 6009 . . . . . . . . . . . . . 14 (𝑝( I ↾ 𝑋)𝑞 ↔ (𝑝𝑋𝑝 I 𝑞))
5 df-br 5149 . . . . . . . . . . . . . 14 (𝑝( I ↾ 𝑋)𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋))
63ideq 5866 . . . . . . . . . . . . . . 15 (𝑝 I 𝑞𝑝 = 𝑞)
76anbi2i 623 . . . . . . . . . . . . . 14 ((𝑝𝑋𝑝 I 𝑞) ↔ (𝑝𝑋𝑝 = 𝑞))
84, 5, 73bitr3i 301 . . . . . . . . . . . . 13 (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) ↔ (𝑝𝑋𝑝 = 𝑞))
98biimpi 216 . . . . . . . . . . . 12 (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) → (𝑝𝑋𝑝 = 𝑞))
109ad2antlr 727 . . . . . . . . . . 11 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝑝𝑋𝑝 = 𝑞))
1110simprd 495 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑝 = 𝑞)
12 df-ov 7434 . . . . . . . . . . 11 (𝑝𝐷𝑝) = (𝐷‘⟨𝑝, 𝑝⟩)
13 opeq2 4879 . . . . . . . . . . . 12 (𝑝 = 𝑞 → ⟨𝑝, 𝑝⟩ = ⟨𝑝, 𝑞⟩)
1413fveq2d 6911 . . . . . . . . . . 11 (𝑝 = 𝑞 → (𝐷‘⟨𝑝, 𝑝⟩) = (𝐷‘⟨𝑝, 𝑞⟩))
1512, 14eqtrid 2787 . . . . . . . . . 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 775 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝐷 ∈ (PsMet‘𝑋))
1810simpld 494 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑝𝑋)
19 psmet0 24334 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑝𝑋) → (𝑝𝐷𝑝) = 0)
2017, 18, 19syl2anc 584 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝑝𝐷𝑝) = 0)
2116, 20eqtr3d 2777 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝐷‘⟨𝑝, 𝑞⟩) = 0)
22 0xr 11306 . . . . . . . . . 10 0 ∈ ℝ*
23 rpxr 13042 . . . . . . . . . 10 (𝑎 ∈ ℝ+𝑎 ∈ ℝ*)
24 rpgt0 13045 . . . . . . . . . 10 (𝑎 ∈ ℝ+ → 0 < 𝑎)
25 lbico1 13438 . . . . . . . . . 10 ((0 ∈ ℝ*𝑎 ∈ ℝ* ∧ 0 < 𝑎) → 0 ∈ (0[,)𝑎))
2622, 23, 24, 25mp3an2i 1465 . . . . . . . . 9 (𝑎 ∈ ℝ+ → 0 ∈ (0[,)𝑎))
2726adantl 481 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 0 ∈ (0[,)𝑎))
2821, 27eqeltrd 2839 . . . . . . 7 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎))
29 psmetf 24332 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
3029ffund 6741 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → Fun 𝐷)
3130ad3antrrr 730 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → Fun 𝐷)
3211, 18eqeltrrd 2840 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑞𝑋)
3318, 32opelxpd 5728 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
3429fdmd 6747 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
3534ad3antrrr 730 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → dom 𝐷 = (𝑋 × 𝑋))
3633, 35eleqtrrd 2842 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ dom 𝐷)
37 fvimacnv 7073 . . . . . . . 8 ((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎))))
3831, 36, 37syl2anc 584 . . . . . . 7 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎))))
3928, 38mpbid 232 . . . . . 6 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎)))
4039adantr 480 . . . . 5 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎)))
41 simpr 484 . . . . 5 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → 𝐴 = (𝐷 “ (0[,)𝑎)))
4240, 41eleqtrrd 2842 . . . 4 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ 𝐴)
43 simplr 769 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → 𝐴𝐹)
44 metust.1 . . . . . . 7 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
4544metustel 24579 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → (𝐴𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎))))
4645ad2antrr 726 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → (𝐴𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎))))
4743, 46mpbid 232 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)))
4842, 47r19.29a 3160 . . 3 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → ⟨𝑝, 𝑞⟩ ∈ 𝐴)
4948ex 412 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) → ⟨𝑝, 𝑞⟩ ∈ 𝐴))
502, 49relssdv 5801 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ( I ↾ 𝑋) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  wss 3963  cop 4637   class class class wbr 5148  cmpt 5231   I cid 5582   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  Rel wrel 5694  Fun wfun 6557  cfv 6563  (class class class)co 7431  0cc0 11153  *cxr 11292   < clt 11293  +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  ax-1cn 11211  ax-addrcl 11214  ax-rnegex 11224  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  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-po 5597  df-so 5598  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-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-rp 13033  df-ico 13390  df-psmet 21374
This theorem is referenced by:  metustfbas  24586  metust  24587
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