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Theorem metidval 32560
Description: Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Assertion
Ref Expression
metidval (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (~Metβ€˜π·) = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (π‘₯𝐷𝑦) = 0)})
Distinct variable groups:   π‘₯,𝑦,𝐷   π‘₯,𝑋,𝑦

Proof of Theorem metidval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 df-metid 32558 . 2 ~Met = (𝑑 ∈ βˆͺ ran PsMet ↦ {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (π‘₯𝑑𝑦) = 0)})
2 simpr 485 . . . . . . . . 9 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ 𝑑 = 𝐷)
32dmeqd 5866 . . . . . . . 8 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ dom 𝑑 = dom 𝐷)
43dmeqd 5866 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ dom dom 𝑑 = dom dom 𝐷)
5 psmetdmdm 23695 . . . . . . . 8 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝑋 = dom dom 𝐷)
65adantr 481 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ 𝑋 = dom dom 𝐷)
74, 6eqtr4d 2774 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ dom dom 𝑑 = 𝑋)
87eleq2d 2818 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ (π‘₯ ∈ dom dom 𝑑 ↔ π‘₯ ∈ 𝑋))
97eleq2d 2818 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ (𝑦 ∈ dom dom 𝑑 ↔ 𝑦 ∈ 𝑋))
108, 9anbi12d 631 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ ((π‘₯ ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ↔ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
112oveqd 7379 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ (π‘₯𝑑𝑦) = (π‘₯𝐷𝑦))
1211eqeq1d 2733 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ ((π‘₯𝑑𝑦) = 0 ↔ (π‘₯𝐷𝑦) = 0))
1310, 12anbi12d 631 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ (((π‘₯ ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (π‘₯𝑑𝑦) = 0) ↔ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (π‘₯𝐷𝑦) = 0)))
1413opabbidv 5176 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (π‘₯𝑑𝑦) = 0)} = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (π‘₯𝐷𝑦) = 0)})
15 elfvunirn 6879 . 2 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝐷 ∈ βˆͺ ran PsMet)
16 opabssxp 5729 . . 3 {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (π‘₯𝐷𝑦) = 0)} βŠ† (𝑋 Γ— 𝑋)
17 elfvex 6885 . . . 4 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝑋 ∈ V)
1817, 17xpexd 7690 . . 3 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝑋 Γ— 𝑋) ∈ V)
19 ssexg 5285 . . 3 (({⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (π‘₯𝐷𝑦) = 0)} βŠ† (𝑋 Γ— 𝑋) ∧ (𝑋 Γ— 𝑋) ∈ V) β†’ {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (π‘₯𝐷𝑦) = 0)} ∈ V)
2016, 18, 19sylancr 587 . 2 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (π‘₯𝐷𝑦) = 0)} ∈ V)
211, 14, 15, 20fvmptd2 6961 1 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (~Metβ€˜π·) = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (π‘₯𝐷𝑦) = 0)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3446   βŠ† wss 3913  βˆͺ cuni 4870  {copab 5172   Γ— cxp 5636  dom cdm 5638  ran crn 5639  β€˜cfv 6501  (class class class)co 7362  0cc0 11060  PsMetcpsmet 20817  ~Metcmetid 32556
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 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-map 8774  df-xr 11202  df-psmet 20825  df-metid 32558
This theorem is referenced by:  metidss  32561  metidv  32562
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