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Theorem ressn2 37914
Description: A class ' R ' restricted to the singleton of the class ' A ' is the ordered pair class abstraction of the class ' A ' and the sets in relation ' R ' to ' A ' (and not in relation to the singleton ' { A } ' ). (Contributed by Peter Mazsa, 16-Jun-2024.)
Assertion
Ref Expression
ressn2 (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴𝐴𝑅𝑢)}
Distinct variable groups:   𝐴,𝑎,𝑢   𝑅,𝑎,𝑢

Proof of Theorem ressn2
StepHypRef Expression
1 dfres2 6045 . 2 (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢)}
2 velsn 4645 . . . . 5 (𝑎 ∈ {𝐴} ↔ 𝑎 = 𝐴)
32anbi1i 623 . . . 4 ((𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴𝑎𝑅𝑢))
4 eqbrb 37703 . . . 4 ((𝑎 = 𝐴𝑎𝑅𝑢) ↔ (𝑎 = 𝐴𝐴𝑅𝑢))
53, 4bitri 275 . . 3 ((𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴𝐴𝑅𝑢))
65opabbii 5215 . 2 {⟨𝑎, 𝑢⟩ ∣ (𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢)} = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴𝐴𝑅𝑢)}
71, 6eqtri 2756 1 (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴𝐴𝑅𝑢)}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1534  wcel 2099  {csn 4629   class class class wbr 5148  {copab 5210  cres 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-res 5690
This theorem is referenced by: (None)
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