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Theorem brabd0 34559
Description: Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.)
Hypotheses
Ref Expression
brabd0.x (𝜑 → ∀𝑥𝜑)
brabd0.y (𝜑 → ∀𝑦𝜑)
brabd0.xch (𝜑 → Ⅎ𝑥𝜒)
brabd0.ych (𝜑 → Ⅎ𝑦𝜒)
brabd0.exa (𝜑𝐴𝑈)
brabd0.exb (𝜑𝐵𝑉)
brabd0.def (𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
brabd0.is ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
brabd0 (𝜑 → (𝐴𝑅𝐵𝜒))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem brabd0
StepHypRef Expression
1 df-br 5031 . . 3 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 brabd0.def . . . 4 (𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
32eleq2d 2875 . . 3 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
41, 3syl5bb 286 . 2 (𝜑 → (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
5 brabd0.x . . 3 (𝜑 → ∀𝑥𝜑)
6 brabd0.y . . 3 (𝜑 → ∀𝑦𝜑)
7 brabd0.xch . . 3 (𝜑 → Ⅎ𝑥𝜒)
8 brabd0.ych . . 3 (𝜑 → Ⅎ𝑦𝜒)
9 brabd0.exa . . 3 (𝜑𝐴𝑈)
10 brabd0.exb . . 3 (𝜑𝐵𝑉)
11 brabd0.is . . 3 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
125, 6, 7, 8, 9, 10, 11opelopabd 34553 . 2 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜒))
134, 12bitrd 282 1 (𝜑 → (𝐴𝑅𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wnf 1785  wcel 2111  cop 4531   class class class wbr 5030  {copab 5092
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 5167  ax-nul 5174  ax-pr 5295
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-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093
This theorem is referenced by:  brabd  34560
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