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Theorem metidss 31310
 Description: As a relation, the metric identification is a subset of a Cartesian product. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Assertion
Ref Expression
metidss (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (𝑋 × 𝑋))

Proof of Theorem metidss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metidval 31309 . 2 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)})
2 opabssxp 5611 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)} ⊆ (𝑋 × 𝑋)
31, 2eqsstrdi 3971 1 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (𝑋 × 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3883  {copab 5096   × cxp 5521  ‘cfv 6332  (class class class)co 7145  0cc0 10544  PsMetcpsmet 20096  ~Metcmetid 31305 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8409  df-xr 10686  df-psmet 20104  df-metid 31307 This theorem is referenced by:  metideq  31312  metider  31313  pstmfval  31315
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