Theorem ressn2 38423

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 6060	. 2 ⊢ (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢)}
2		velsn 4646	. . . . 5 ⊢ (𝑎 ∈ {𝐴} ↔ 𝑎 = 𝐴)
3	2	anbi1i 624	. . . 4 ⊢ ((𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴 ∧ 𝑎𝑅𝑢))
4		eqbrb 38213	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢))
5	3, 4	bitri 275	. . 3 ⊢ ((𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢))
6	5	opabbii 5214	. 2 ⊢ {⟨𝑎, 𝑢⟩ ∣ (𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢)} = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢)}
7	1, 6	eqtri 2762	1 ⊢ (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢)}

Syntax hints:   ∧ wa 395   = wceq 1536   ∈ wcel 2105  {csn 4630   class class class wbr 5147  {copab 5209   ↾ cres 5690

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-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-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-res 5700

This theorem is referenced by: (None)