Theorem ressn2 38853

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 = 𝐴 ∧ 𝐴𝑅𝑢)}

Distinct variable groups:   𝐴,𝑎,𝑢   𝑅,𝑎,𝑢

Proof of Theorem ressn2

Step	Hyp	Ref	Expression
1		dfres2 6006	. 2 ⊢ (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢)}
2		velsn 4583	. . . . 5 ⊢ (𝑎 ∈ {𝐴} ↔ 𝑎 = 𝐴)
3	2	anbi1i 625	. . . 4 ⊢ ((𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴 ∧ 𝑎𝑅𝑢))
4		eqbrb 38560	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢))
5	3, 4	bitri 275	. . 3 ⊢ ((𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢))
6	5	opabbii 5152	. 2 ⊢ {⟨𝑎, 𝑢⟩ ∣ (𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢)} = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢)}
7	1, 6	eqtri 2759	1 ⊢ (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢)}

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {csn 4567   class class class wbr 5085  {copab 5147   ↾ cres 5633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-res 5643

This theorem is referenced by: (None)