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Theorem brabd 35347
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 1909 . 2 (𝜑 → ∀𝑥𝜑)
2 ax-5 1909 . 2 (𝜑 → ∀𝑦𝜑)
3 nfvd 1914 . 2 (𝜑 → Ⅎ𝑥𝜒)
4 nfvd 1914 . 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 35346 1 (𝜑 → (𝐴𝑅𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1537  wcel 2101   class class class wbr 5077  {copab 5139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pr 5355
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-br 5078  df-opab 5140
This theorem is referenced by:  bj-imdirval3  35383
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