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Theorem brabd 37130
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 1907 . 2 (𝜑 → ∀𝑥𝜑)
2 ax-5 1907 . 2 (𝜑 → ∀𝑦𝜑)
3 nfvd 1912 . 2 (𝜑 → Ⅎ𝑥𝜒)
4 nfvd 1912 . 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 37129 1 (𝜑 → (𝐴𝑅𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105   class class class wbr 5147  {copab 5209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210
This theorem is referenced by:  bj-imdirval3  37166
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