Theorem brcoss 38474

Description: 𝐴 and 𝐵 are cosets by 𝑅: a binary relation. (Contributed by Peter Mazsa, 27-Dec-2018.)

Assertion

Ref	Expression
brcoss	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))

Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝑅   𝑢,𝑉   𝑢,𝑊

Proof of Theorem brcoss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5095	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑢𝑅𝑥 ↔ 𝑢𝑅𝐴))
2		breq2 5095	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑢𝑅𝑦 ↔ 𝑢𝑅𝐵))
3	1, 2	bi2anan9 638	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ (𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))
4	3	exbidv 1922	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))
5		df-coss 38454	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
6	4, 5	brabga 5474	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111   class class class wbr 5091   ≀ ccoss 38221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-coss 38454

This theorem is referenced by:  brcoss2  38475  brcoss3  38476  brcosscnvcoss  38477  cocossss  38479  br1cossres  38482  eldmcoss2  38502  brcosscnv  38515  trcoss  38525