Theorem brcoss 38387

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 5170	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑢𝑅𝑥 ↔ 𝑢𝑅𝐴))
2		breq2 5170	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑢𝑅𝑦 ↔ 𝑢𝑅𝐵))
3	1, 2	bi2anan9 637	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ (𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))
4	3	exbidv 1920	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))
5		df-coss 38367	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
6	4, 5	brabga 5553	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108   class class class wbr 5166   ≀ ccoss 38135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-coss 38367

This theorem is referenced by:  brcoss2  38388  brcoss3  38389  brcosscnvcoss  38390  cocossss  38392  br1cossres  38395  eldmcoss2  38415  brcosscnv  38428  trcoss  38438