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Theorem brabd 34940
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
brabd.exa (𝜑𝐴𝑈)
brabd.exb (𝜑𝐵𝑉)
brabd.def (𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
brabd.is ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
brabd (𝜑 → (𝐴𝑅𝐵𝜒))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦   𝜒,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem brabd
StepHypRef Expression
1 ax-5 1917 . 2 (𝜑 → ∀𝑥𝜑)
2 ax-5 1917 . 2 (𝜑 → ∀𝑦𝜑)
3 nfvd 1922 . 2 (𝜑 → Ⅎ𝑥𝜒)
4 nfvd 1922 . 2 (𝜑 → Ⅎ𝑦𝜒)
5 brabd.exa . 2 (𝜑𝐴𝑈)
6 brabd.exb . 2 (𝜑𝐵𝑉)
7 brabd.def . 2 (𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
8 brabd.is . 2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
91, 2, 3, 4, 5, 6, 7, 8brabd0 34939 1 (𝜑 → (𝐴𝑅𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114   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 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-dif 3846  df-un 3848  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093
This theorem is referenced by:  bj-imdirval3  34976
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