Theorem brressn 38442

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 6004	. . 3 ⊢ (𝐶 ∈ 𝑊 → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 ∈ {𝐴} ∧ 𝐵𝑅𝐶)))
2	1	adantl 481	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 ∈ {𝐴} ∧ 𝐵𝑅𝐶)))
3		elsng 4640	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ {𝐴} ↔ 𝐵 = 𝐴))
4	3	adantr 480	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∈ {𝐴} ↔ 𝐵 = 𝐴))
5	4	anbi1d 631	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐵 ∈ {𝐴} ∧ 𝐵𝑅𝐶) ↔ (𝐵 = 𝐴 ∧ 𝐵𝑅𝐶)))
6	2, 5	bitrd 279	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 = 𝐴 ∧ 𝐵𝑅𝐶)))

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