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Theorem metidval 30131
Description: Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Assertion
Ref Expression
metidval (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)})
Distinct variable groups:   𝑥,𝑦,𝐷   𝑥,𝑋,𝑦

Proof of Theorem metidval
Dummy variables 𝑤 𝑑 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-metid 30129 . . 3 ~Met = (𝑑 ran PsMet ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)})
21a1i 11 . 2 (𝐷 ∈ (PsMet‘𝑋) → ~Met = (𝑑 ran PsMet ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)}))
3 simpr 479 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
43dmeqd 5401 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom 𝑑 = dom 𝐷)
54dmeqd 5401 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = dom dom 𝐷)
6 psmetdmdm 22200 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
76adantr 472 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝐷)
85, 7eqtr4d 2729 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
98eleq2d 2757 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥 ∈ dom dom 𝑑𝑥𝑋))
108eleq2d 2757 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑦 ∈ dom dom 𝑑𝑦𝑋))
119, 10anbi12d 749 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑥 ∈ dom dom 𝑑𝑦 ∈ dom dom 𝑑) ↔ (𝑥𝑋𝑦𝑋)))
123oveqd 6750 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
1312eqeq1d 2694 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑥𝑑𝑦) = 0 ↔ (𝑥𝐷𝑦) = 0))
1411, 13anbi12d 749 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (((𝑥 ∈ dom dom 𝑑𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0) ↔ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)))
1514opabbidv 4792 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)})
16 elfvdm 6301 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
17 fveq2 6272 . . . . . 6 (𝑥 = 𝑋 → (PsMet‘𝑥) = (PsMet‘𝑋))
1817eleq2d 2757 . . . . 5 (𝑥 = 𝑋 → (𝐷 ∈ (PsMet‘𝑥) ↔ 𝐷 ∈ (PsMet‘𝑋)))
1918rspcev 3381 . . . 4 ((𝑋 ∈ dom PsMet ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
2016, 19mpancom 706 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
21 df-psmet 19829 . . . . 5 PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧𝑥𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
2221funmpt2 6008 . . . 4 Fun PsMet
23 elunirn 6592 . . . 4 (Fun PsMet → (𝐷 ran PsMet ↔ ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥)))
2422, 23ax-mp 5 . . 3 (𝐷 ran PsMet ↔ ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
2520, 24sylibr 224 . 2 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ran PsMet)
26 opabssxp 5270 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)} ⊆ (𝑋 × 𝑋)
27 elfvex 6302 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
28 xpexg 7045 . . . 4 ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑋 × 𝑋) ∈ V)
2927, 27, 28syl2anc 696 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝑋 × 𝑋) ∈ V)
30 ssexg 4880 . . 3 (({⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)} ⊆ (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ V) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)} ∈ V)
3126, 29, 30sylancr 698 . 2 (𝐷 ∈ (PsMet‘𝑋) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)} ∈ V)
322, 15, 25, 31fvmptd 6370 1 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1564  wcel 2071  wral 2982  wrex 2983  {crab 2986  Vcvv 3272  wss 3648   cuni 4512   class class class wbr 4728  {copab 4788  cmpt 4805   × cxp 5184  dom cdm 5186  ran crn 5187  Fun wfun 5963  cfv 5969  (class class class)co 6733  𝑚 cmap 7942  0cc0 10017  *cxr 10154  cle 10156   +𝑒 cxad 12026  PsMetcpsmet 19821  ~Metcmetid 30127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-8 2073  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-sep 4857  ax-nul 4865  ax-pow 4916  ax-pr 4979  ax-un 7034  ax-cnex 10073  ax-resscn 10074
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-rex 2988  df-rab 2991  df-v 3274  df-sbc 3510  df-csb 3608  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-nul 3992  df-if 4163  df-pw 4236  df-sn 4254  df-pr 4256  df-op 4260  df-uni 4513  df-br 4729  df-opab 4789  df-mpt 4806  df-id 5096  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-rn 5197  df-iota 5932  df-fun 5971  df-fn 5972  df-f 5973  df-fv 5977  df-ov 6736  df-oprab 6737  df-mpt2 6738  df-map 7944  df-xr 10159  df-psmet 19829  df-metid 30129
This theorem is referenced by:  metidss  30132  metidv  30133
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