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Theorem brressn 38422
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 6006 . . 3 (𝐶𝑊 → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 ∈ {𝐴} ∧ 𝐵𝑅𝐶)))
21adantl 481 . 2 ((𝐵𝑉𝐶𝑊) → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 ∈ {𝐴} ∧ 𝐵𝑅𝐶)))
3 elsng 4644 . . . 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 1536  wcel 2105  {csn 4630   class class class wbr 5147  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-res 5700
This theorem is referenced by:  refressn  38424  antisymressn  38425  trressn  38426
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