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Theorem ressn2 37312
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 6042 . 2 (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢)}
2 velsn 4645 . . . . 5 (𝑎 ∈ {𝐴} ↔ 𝑎 = 𝐴)
32anbi1i 625 . . . 4 ((𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴𝑎𝑅𝑢))
4 eqbrb 37099 . . . 4 ((𝑎 = 𝐴𝑎𝑅𝑢) ↔ (𝑎 = 𝐴𝐴𝑅𝑢))
53, 4bitri 275 . . 3 ((𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴𝐴𝑅𝑢))
65opabbii 5216 . 2 {⟨𝑎, 𝑢⟩ ∣ (𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢)} = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴𝐴𝑅𝑢)}
71, 6eqtri 2761 1 (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴𝐴𝑅𝑢)}
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  {csn 4629   class class class wbr 5149  {copab 5211  cres 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-res 5689
This theorem is referenced by: (None)
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