Theorem metidval 32560

Description: Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)

Assertion

Ref	Expression
metidval	⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑥𝐷𝑦) = 0)})

Distinct variable groups:   𝑥,𝑦,𝐷   𝑥,𝑋,𝑦

Proof of Theorem metidval

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-metid 32558	. 2 ⊢ ~Met = (𝑑 ∈ ∪ ran PsMet ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)})
2		simpr 485	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
3	2	dmeqd 5866	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom 𝑑 = dom 𝐷)
4	3	dmeqd 5866	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = dom dom 𝐷)
5		psmetdmdm 23695	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
6	5	adantr 481	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝐷)
7	4, 6	eqtr4d 2774	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
8	7	eleq2d 2818	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥 ∈ dom dom 𝑑 ↔ 𝑥 ∈ 𝑋))
9	7	eleq2d 2818	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑦 ∈ dom dom 𝑑 ↔ 𝑦 ∈ 𝑋))
10	8, 9	anbi12d 631	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑥 ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
11	2	oveqd 7379	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
12	11	eqeq1d 2733	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑥𝑑𝑦) = 0 ↔ (𝑥𝐷𝑦) = 0))
13	10, 12	anbi12d 631	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (((𝑥 ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑥𝐷𝑦) = 0)))
14	13	opabbidv 5176	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑥𝐷𝑦) = 0)})
15		elfvunirn 6879	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ ∪ ran PsMet)
16		opabssxp 5729	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑥𝐷𝑦) = 0)} ⊆ (𝑋 × 𝑋)
17		elfvex 6885	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
18	17, 17	xpexd 7690	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑋 × 𝑋) ∈ V)
19		ssexg 5285	. . 3 ⊢ (({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑥𝐷𝑦) = 0)} ⊆ (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ V) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑥𝐷𝑦) = 0)} ∈ V)
20	16, 18, 19	sylancr 587	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑥𝐷𝑦) = 0)} ∈ V)
21	1, 14, 15, 20	fvmptd2 6961	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑥𝐷𝑦) = 0)})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3446   ⊆ wss 3913  ∪ cuni 4870  {copab 5172   × cxp 5636  dom cdm 5638  ran crn 5639  ‘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