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Theorem brressn 37300
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 5987 . . 3 (𝐶𝑊 → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 ∈ {𝐴} ∧ 𝐵𝑅𝐶)))
21adantl 483 . 2 ((𝐵𝑉𝐶𝑊) → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 ∈ {𝐴} ∧ 𝐵𝑅𝐶)))
3 elsng 4642 . . . 4 (𝐵𝑉 → (𝐵 ∈ {𝐴} ↔ 𝐵 = 𝐴))
43adantr 482 . . 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 205  wa 397   = wceq 1542  wcel 2107  {csn 4628   class class class wbr 5148  cres 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-res 5688
This theorem is referenced by:  refressn  37302  antisymressn  37303  trressn  37304
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