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Theorem ressn2 37823
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 6034 . 2 (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢)}
2 velsn 4639 . . . . 5 (𝑎 ∈ {𝐴} ↔ 𝑎 = 𝐴)
32anbi1i 623 . . . 4 ((𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴𝑎𝑅𝑢))
4 eqbrb 37610 . . . 4 ((𝑎 = 𝐴𝑎𝑅𝑢) ↔ (𝑎 = 𝐴𝐴𝑅𝑢))
53, 4bitri 275 . . 3 ((𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴𝐴𝑅𝑢))
65opabbii 5208 . 2 {⟨𝑎, 𝑢⟩ ∣ (𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢)} = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴𝐴𝑅𝑢)}
71, 6eqtri 2754 1 (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴𝐴𝑅𝑢)}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1533  wcel 2098  {csn 4623   class class class wbr 5141  {copab 5203  cres 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-res 5681
This theorem is referenced by: (None)
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