Theorem brressn 39383

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 5973	. . 3 ⊢ (𝐶 ∈ 𝑊 → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 ∈ {𝐴} ∧ 𝐵𝑅𝐶)))
2	1	adantl 487	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 ∈ {𝐴} ∧ 𝐵𝑅𝐶)))
3		elsng 4597	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ {𝐴} ↔ 𝐵 = 𝐴))
4	3	adantr 486	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∈ {𝐴} ↔ 𝐵 = 𝐴))
5	4	anbi1d 643	. 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 401   = wceq 1570   ∈ wcel 2145  {csn 4583   class class class wbr 5102   ↾ cres 5649

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-res 5659

This theorem is used by:  refressn  39385  antisymressn  39386  trressn  39387