Theorem brressn 39208

Description: Binary relation on a restriction to a singleton. (Contributed by Peter Mazsa, 11-Jun-2024.)

Assertion

Ref	Expression
brressn	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 = 𝐴 ∧ 𝐵𝑅𝐶)))

Proof of Theorem brressn

Step	Hyp	Ref	Expression
1		brres 5984	. . 3 ⊢ (𝐶 ∈ 𝑊 → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 ∈ {𝐴} ∧ 𝐵𝑅𝐶)))
2	1	adantl 486	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 ∈ {𝐴} ∧ 𝐵𝑅𝐶)))
3		elsng 4602	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ {𝐴} ↔ 𝐵 = 𝐴))
4	3	adantr 485	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∈ {𝐴} ↔ 𝐵 = 𝐴))
5	4	anbi1d 642	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐵 ∈ {𝐴} ∧ 𝐵𝑅𝐶) ↔ (𝐵 = 𝐴 ∧ 𝐵𝑅𝐶)))
6	2, 5	bitrd 282	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 = 𝐴 ∧ 𝐵𝑅𝐶)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {csn 4588   class class class wbr 5108   ↾ cres 5662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-res 5672

This theorem is used by:  refressn  39210  antisymressn  39211  trressn  39212