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Theorem brressn 38442
Description: Binary relation on a restriction to a singleton. (Contributed by Peter Mazsa, 11-Jun-2024.)
Assertion
Ref Expression
brressn ((𝐵𝑉𝐶𝑊) → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 = 𝐴𝐵𝑅𝐶)))

Proof of Theorem brressn
StepHypRef Expression
1 brres 6004 . . 3 (𝐶𝑊 → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 ∈ {𝐴} ∧ 𝐵𝑅𝐶)))
21adantl 481 . 2 ((𝐵𝑉𝐶𝑊) → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 ∈ {𝐴} ∧ 𝐵𝑅𝐶)))
3 elsng 4640 . . . 4 (𝐵𝑉 → (𝐵 ∈ {𝐴} ↔ 𝐵 = 𝐴))
43adantr 480 . . 3 ((𝐵𝑉𝐶𝑊) → (𝐵 ∈ {𝐴} ↔ 𝐵 = 𝐴))
54anbi1d 631 . 2 ((𝐵𝑉𝐶𝑊) → ((𝐵 ∈ {𝐴} ∧ 𝐵𝑅𝐶) ↔ (𝐵 = 𝐴𝐵𝑅𝐶)))
62, 5bitrd 279 1 ((𝐵𝑉𝐶𝑊) → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 = 𝐴𝐵𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {csn 4626   class class class wbr 5143  cres 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-res 5697
This theorem is referenced by:  refressn  38444  antisymressn  38445  trressn  38446
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