Theorem metidv 34260

Description: 𝐴 and 𝐵 identify by the metric 𝐷 if their distance is zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)

Assertion

Ref	Expression
metidv	⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))

Proof of Theorem metidv

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2851	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ 𝑋 ↔ 𝐴 ∈ 𝑋))
2		eleq1 2851	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋))
3	1, 2	bi2anan9 649	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)))
4		oveq12 7421	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎𝐷𝑏) = (𝐴𝐷𝐵))
5	4	eqeq1d 2765	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎𝐷𝑏) = 0 ↔ (𝐴𝐷𝐵) = 0))
6	3, 5	anbi12d 643	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0)))
7		eqid 2763	. . . 4 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)}
8	6, 7	brabga 5520	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵 ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0)))
9	8	adantl 486	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵 ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0)))
10		metidval 34258	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)})
11	10	adantr 485	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (~Met‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)})
12	11	breqd 5121	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐵 ↔ 𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ (𝑎𝐷𝑏) = 0)}𝐵))
13		ibar 537	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0)))
14	13	adantl 486	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵) = 0 ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐴𝐷𝐵) = 0)))
15	9, 12, 14	3bitr4d 314	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   class class class wbr 5110  {copab 5174  ‘cfv 6538  (class class class)co 7412  0cc0 11101  PsMetcpsmet 21487  ~_Metcmetid 34254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8827  df-xr 11248  df-psmet 21495  df-metid 34256

This theorem is referenced by:  metideq  34261  metider  34262  pstmfval  34264  pstmxmet  34265