Theorem ressn2 38398

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 6070	. 2 ⊢ (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢)}
2		velsn 4664	. . . . 5 ⊢ (𝑎 ∈ {𝐴} ↔ 𝑎 = 𝐴)
3	2	anbi1i 623	. . . 4 ⊢ ((𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴 ∧ 𝑎𝑅𝑢))
4		eqbrb 38188	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢))
5	3, 4	bitri 275	. . 3 ⊢ ((𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢))
6	5	opabbii 5233	. 2 ⊢ {⟨𝑎, 𝑢⟩ ∣ (𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢)} = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢)}
7	1, 6	eqtri 2768	1 ⊢ (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢)}

Syntax hints:   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {csn 4648   class class class wbr 5166  {copab 5228   ↾ cres 5702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-res 5712

This theorem is referenced by: (None)