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Theorem metidval 30242
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 30240 . . 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 5481 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom 𝑑 = dom 𝐷)
54dmeqd 5481 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = dom dom 𝐷)
6 psmetdmdm 22311 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
76adantr 472 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝐷)
85, 7eqtr4d 2797 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
98eleq2d 2825 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥 ∈ dom dom 𝑑𝑥𝑋))
108eleq2d 2825 . . . . 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 6830 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
1312eqeq1d 2762 . . . 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 4868 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)})
16 elfvdm 6381 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
17 fveq2 6352 . . . . . 6 (𝑥 = 𝑋 → (PsMet‘𝑥) = (PsMet‘𝑋))
1817eleq2d 2825 . . . . 5 (𝑥 = 𝑋 → (𝐷 ∈ (PsMet‘𝑥) ↔ 𝐷 ∈ (PsMet‘𝑋)))
1918rspcev 3449 . . . 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 19940 . . . . 5 PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧𝑥𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
2221funmpt2 6088 . . . 4 Fun PsMet
23 elunirn 6672 . . . 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 5350 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)} ⊆ (𝑋 × 𝑋)
27 elfvex 6382 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
28 xpexg 7125 . . . 4 ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑋 × 𝑋) ∈ V)
2927, 27, 28syl2anc 696 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝑋 × 𝑋) ∈ V)
30 ssexg 4956 . . 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 6450 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 1632  wcel 2139  wral 3050  wrex 3051  {crab 3054  Vcvv 3340  wss 3715   cuni 4588   class class class wbr 4804  {copab 4864  cmpt 4881   × cxp 5264  dom cdm 5266  ran crn 5267  Fun wfun 6043  cfv 6049  (class class class)co 6813  𝑚 cmap 8023  0cc0 10128  *cxr 10265  cle 10267   +𝑒 cxad 12137  PsMetcpsmet 19932  ~Metcmetid 30238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025  df-xr 10270  df-psmet 19940  df-metid 30240
This theorem is referenced by:  metidss  30243  metidv  30244
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