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Theorem metidss 32536
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 ∈ (PsMetβ€˜π‘‹) β†’ (~Metβ€˜π·) βŠ† (𝑋 Γ— 𝑋))

Proof of Theorem metidss
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metidval 32535 . 2 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (~Metβ€˜π·) = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (π‘₯𝐷𝑦) = 0)})
2 opabssxp 5728 . 2 {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (π‘₯𝐷𝑦) = 0)} βŠ† (𝑋 Γ— 𝑋)
31, 2eqsstrdi 4002 1 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (~Metβ€˜π·) βŠ† (𝑋 Γ— 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3914  {copab 5171   Γ— cxp 5635  β€˜cfv 6500  (class class class)co 7361  0cc0 11059  PsMetcpsmet 20803  ~Metcmetid 32531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773  df-xr 11201  df-psmet 20811  df-metid 32533
This theorem is referenced by:  metideq  32538  metider  32539  pstmfval  32541
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