Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  metidval Structured version   Visualization version   GIF version

Theorem metidval 33851
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 variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 df-metid 33849 . 2 ~Met = (𝑑 ran PsMet ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)})
2 simpr 484 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
32dmeqd 5919 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom 𝑑 = dom 𝐷)
43dmeqd 5919 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = dom dom 𝐷)
5 psmetdmdm 24331 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
65adantr 480 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝐷)
74, 6eqtr4d 2778 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
87eleq2d 2825 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥 ∈ dom dom 𝑑𝑥𝑋))
97eleq2d 2825 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑦 ∈ dom dom 𝑑𝑦𝑋))
108, 9anbi12d 632 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑥 ∈ dom dom 𝑑𝑦 ∈ dom dom 𝑑) ↔ (𝑥𝑋𝑦𝑋)))
112oveqd 7448 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
1211eqeq1d 2737 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑥𝑑𝑦) = 0 ↔ (𝑥𝐷𝑦) = 0))
1310, 12anbi12d 632 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (((𝑥 ∈ dom dom 𝑑𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0) ↔ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)))
1413opabbidv 5214 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)})
15 elfvunirn 6939 . 2 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ran PsMet)
16 opabssxp 5781 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)} ⊆ (𝑋 × 𝑋)
17 elfvex 6945 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
1817, 17xpexd 7770 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝑋 × 𝑋) ∈ V)
19 ssexg 5329 . . 3 (({⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)} ⊆ (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ V) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)} ∈ V)
2016, 18, 19sylancr 587 . 2 (𝐷 ∈ (PsMet‘𝑋) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)} ∈ V)
211, 14, 15, 20fvmptd2 7024 1 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963   cuni 4912  {copab 5210   × cxp 5687  dom cdm 5689  ran crn 5690  cfv 6563  (class class class)co 7431  0cc0 11153  PsMetcpsmet 21366  ~Metcmetid 33847
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-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-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-xr 11297  df-psmet 21374  df-metid 33849
This theorem is referenced by:  metidss  33852  metidv  33853
  Copyright terms: Public domain W3C validator