Theorem br1cossres2 38422

Description: 𝐵 and 𝐶 are cosets by a restriction: a binary relation. (Contributed by Peter Mazsa, 3-Jan-2018.)

Assertion

Ref	Expression
br1cossres2	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ↾ 𝐴)𝐶 ↔ ∃𝑥 ∈ 𝐴 (𝐵 ∈ [𝑥]𝑅 ∧ 𝐶 ∈ [𝑥]𝑅)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝑅   𝑥,𝑉   𝑥,𝑊

Proof of Theorem br1cossres2

Step	Hyp	Ref	Expression
1		br1cossres 38421	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ↾ 𝐴)𝐶 ↔ ∃𝑥 ∈ 𝐴 (𝑥𝑅𝐵 ∧ 𝑥𝑅𝐶)))
2		exanres3 38278	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑥 ∈ 𝐴 (𝐵 ∈ [𝑥]𝑅 ∧ 𝐶 ∈ [𝑥]𝑅) ↔ ∃𝑥 ∈ 𝐴 (𝑥𝑅𝐵 ∧ 𝑥𝑅𝐶)))
3	1, 2	bitr4d 282	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ↾ 𝐴)𝐶 ↔ ∃𝑥 ∈ 𝐴 (𝐵 ∈ [𝑥]𝑅 ∧ 𝐶 ∈ [𝑥]𝑅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2106  ∃wrex 3068   class class class wbr 5148   ↾ cres 5691  [cec 8742   ≀ ccoss 38162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-coss 38393

This theorem is referenced by:  relbrcoss  38428