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Theorem metustid 23308
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 5855 . . 3 Rel ( I ↾ 𝑋)
21a1i 11 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → Rel ( I ↾ 𝑋))
3 vex 3402 . . . . . . . . . . . . . . 15 𝑞 ∈ V
43brresi 5835 . . . . . . . . . . . . . 14 (𝑝( I ↾ 𝑋)𝑞 ↔ (𝑝𝑋𝑝 I 𝑞))
5 df-br 5032 . . . . . . . . . . . . . 14 (𝑝( I ↾ 𝑋)𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋))
63ideq 5696 . . . . . . . . . . . . . . 15 (𝑝 I 𝑞𝑝 = 𝑞)
76anbi2i 626 . . . . . . . . . . . . . 14 ((𝑝𝑋𝑝 I 𝑞) ↔ (𝑝𝑋𝑝 = 𝑞))
84, 5, 73bitr3i 304 . . . . . . . . . . . . 13 (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) ↔ (𝑝𝑋𝑝 = 𝑞))
98biimpi 219 . . . . . . . . . . . 12 (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) → (𝑝𝑋𝑝 = 𝑞))
109ad2antlr 727 . . . . . . . . . . 11 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝑝𝑋𝑝 = 𝑞))
1110simprd 499 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑝 = 𝑞)
12 df-ov 7174 . . . . . . . . . . 11 (𝑝𝐷𝑝) = (𝐷‘⟨𝑝, 𝑝⟩)
13 opeq2 4762 . . . . . . . . . . . 12 (𝑝 = 𝑞 → ⟨𝑝, 𝑝⟩ = ⟨𝑝, 𝑞⟩)
1413fveq2d 6679 . . . . . . . . . . 11 (𝑝 = 𝑞 → (𝐷‘⟨𝑝, 𝑝⟩) = (𝐷‘⟨𝑝, 𝑞⟩))
1512, 14syl5eq 2785 . . . . . . . . . 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 498 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑝𝑋)
19 psmet0 23062 . . . . . . . . . 10 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑝𝑋) → (𝑝𝐷𝑝) = 0)
2017, 18, 19syl2anc 587 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝑝𝐷𝑝) = 0)
2116, 20eqtr3d 2775 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝐷‘⟨𝑝, 𝑞⟩) = 0)
22 0xr 10767 . . . . . . . . . 10 0 ∈ ℝ*
23 rpxr 12482 . . . . . . . . . 10 (𝑎 ∈ ℝ+𝑎 ∈ ℝ*)
24 rpgt0 12485 . . . . . . . . . 10 (𝑎 ∈ ℝ+ → 0 < 𝑎)
25 lbico1 12876 . . . . . . . . . 10 ((0 ∈ ℝ*𝑎 ∈ ℝ* ∧ 0 < 𝑎) → 0 ∈ (0[,)𝑎))
2622, 23, 24, 25mp3an2i 1467 . . . . . . . . 9 (𝑎 ∈ ℝ+ → 0 ∈ (0[,)𝑎))
2726adantl 485 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 0 ∈ (0[,)𝑎))
2821, 27eqeltrd 2833 . . . . . . 7 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎))
29 psmetf 23060 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
3029ffund 6509 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → Fun 𝐷)
3130ad3antrrr 730 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → Fun 𝐷)
3211, 18eqeltrrd 2834 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑞𝑋)
3318, 32opelxpd 5564 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
3429fdmd 6516 . . . . . . . . . 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 2836 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ dom 𝐷)
37 fvimacnv 6831 . . . . . . . 8 ((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎))))
3831, 36, 37syl2anc 587 . . . . . . 7 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎))))
3928, 38mpbid 235 . . . . . 6 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎)))
4039adantr 484 . . . . 5 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎)))
41 simpr 488 . . . . 5 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → 𝐴 = (𝐷 “ (0[,)𝑎)))
4240, 41eleqtrrd 2836 . . . 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 23304 . . . . . 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 235 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)))
4842, 47r19.29a 3199 . . 3 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → ⟨𝑝, 𝑞⟩ ∈ 𝐴)
4948ex 416 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) → ⟨𝑝, 𝑞⟩ ∈ 𝐴))
502, 49relssdv 5633 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ( I ↾ 𝑋) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2113  wrex 3054  wss 3844  cop 4523   class class class wbr 5031  cmpt 5111   I cid 5429   × cxp 5524  ccnv 5525  dom cdm 5526  ran crn 5527  cres 5528  cima 5529  Rel wrel 5531  Fun wfun 6334  cfv 6340  (class class class)co 7171  0cc0 10616  *cxr 10753   < clt 10754  +crp 12473  [,)cico 12824  PsMetcpsmet 20202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7480  ax-cnex 10672  ax-resscn 10673  ax-1cn 10674  ax-addrcl 10677  ax-rnegex 10687  ax-cnre 10689  ax-pre-lttri 10690  ax-pre-lttrn 10691
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3683  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-po 5443  df-so 5444  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7174  df-oprab 7175  df-mpo 7176  df-er 8321  df-map 8440  df-en 8557  df-dom 8558  df-sdom 8559  df-pnf 10756  df-mnf 10757  df-xr 10758  df-ltxr 10759  df-le 10760  df-rp 12474  df-ico 12828  df-psmet 20210
This theorem is referenced by:  metustfbas  23311  metust  23312
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