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Theorem metustid 23910
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 5966 . . 3 Rel ( I ↾ 𝑋)
21a1i 11 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → Rel ( I ↾ 𝑋))
3 vex 3449 . . . . . . . . . . . . . . 15 𝑞 ∈ V
43brresi 5946 . . . . . . . . . . . . . 14 (𝑝( I ↾ 𝑋)𝑞 ↔ (𝑝𝑋𝑝 I 𝑞))
5 df-br 5106 . . . . . . . . . . . . . 14 (𝑝( I ↾ 𝑋)𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋))
63ideq 5808 . . . . . . . . . . . . . . 15 (𝑝 I 𝑞𝑝 = 𝑞)
76anbi2i 623 . . . . . . . . . . . . . 14 ((𝑝𝑋𝑝 I 𝑞) ↔ (𝑝𝑋𝑝 = 𝑞))
84, 5, 73bitr3i 300 . . . . . . . . . . . . 13 (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) ↔ (𝑝𝑋𝑝 = 𝑞))
98biimpi 215 . . . . . . . . . . . 12 (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) → (𝑝𝑋𝑝 = 𝑞))
109ad2antlr 725 . . . . . . . . . . 11 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝑝𝑋𝑝 = 𝑞))
1110simprd 496 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑝 = 𝑞)
12 df-ov 7360 . . . . . . . . . . 11 (𝑝𝐷𝑝) = (𝐷‘⟨𝑝, 𝑝⟩)
13 opeq2 4831 . . . . . . . . . . . 12 (𝑝 = 𝑞 → ⟨𝑝, 𝑝⟩ = ⟨𝑝, 𝑞⟩)
1413fveq2d 6846 . . . . . . . . . . 11 (𝑝 = 𝑞 → (𝐷‘⟨𝑝, 𝑝⟩) = (𝐷‘⟨𝑝, 𝑞⟩))
1512, 14eqtrid 2788 . . . . . . . . . 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 773 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝐷 ∈ (PsMet‘𝑋))
1810simpld 495 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑝𝑋)
19 psmet0 23661 . . . . . . . . . 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 2778 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝐷‘⟨𝑝, 𝑞⟩) = 0)
22 0xr 11202 . . . . . . . . . 10 0 ∈ ℝ*
23 rpxr 12924 . . . . . . . . . 10 (𝑎 ∈ ℝ+𝑎 ∈ ℝ*)
24 rpgt0 12927 . . . . . . . . . 10 (𝑎 ∈ ℝ+ → 0 < 𝑎)
25 lbico1 13318 . . . . . . . . . 10 ((0 ∈ ℝ*𝑎 ∈ ℝ* ∧ 0 < 𝑎) → 0 ∈ (0[,)𝑎))
2622, 23, 24, 25mp3an2i 1466 . . . . . . . . 9 (𝑎 ∈ ℝ+ → 0 ∈ (0[,)𝑎))
2726adantl 482 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 0 ∈ (0[,)𝑎))
2821, 27eqeltrd 2838 . . . . . . 7 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎))
29 psmetf 23659 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
3029ffund 6672 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → Fun 𝐷)
3130ad3antrrr 728 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → Fun 𝐷)
3211, 18eqeltrrd 2839 . . . . . . . . . 10 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → 𝑞𝑋)
3318, 32opelxpd 5671 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
3429fdmd 6679 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
3534ad3antrrr 728 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → dom 𝐷 = (𝑋 × 𝑋))
3633, 35eleqtrrd 2841 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ dom 𝐷)
37 fvimacnv 7003 . . . . . . . 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 231 . . . . . 6 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) → ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎)))
4039adantr 481 . . . . 5 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎)))
41 simpr 485 . . . . 5 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → 𝐴 = (𝐷 “ (0[,)𝑎)))
4240, 41eleqtrrd 2841 . . . 4 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ⟨𝑝, 𝑞⟩ ∈ 𝐴)
43 simplr 767 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → 𝐴𝐹)
44 metust.1 . . . . . . 7 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
4544metustel 23906 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → (𝐴𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎))))
4645ad2antrr 724 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → (𝐴𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎))))
4743, 46mpbid 231 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)))
4842, 47r19.29a 3159 . . 3 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) ∧ ⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋)) → ⟨𝑝, 𝑞⟩ ∈ 𝐴)
4948ex 413 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → (⟨𝑝, 𝑞⟩ ∈ ( I ↾ 𝑋) → ⟨𝑝, 𝑞⟩ ∈ 𝐴))
502, 49relssdv 5744 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ( I ↾ 𝑋) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3073  wss 3910  cop 4592   class class class wbr 5105  cmpt 5188   I cid 5530   × cxp 5631  ccnv 5632  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  Rel wrel 5638  Fun wfun 6490  cfv 6496  (class class class)co 7357  0cc0 11051  *cxr 11188   < clt 11189  +crp 12915  [,)cico 13266  PsMetcpsmet 20780
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 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-addrcl 11112  ax-rnegex 11122  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  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 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-rp 12916  df-ico 13270  df-psmet 20788
This theorem is referenced by:  metustfbas  23913  metust  23914
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