Theorem br1cossres2 39034

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 39033	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ↾ 𝐴)𝐶 ↔ ∃𝑥 ∈ 𝐴 (𝑥𝑅𝐵 ∧ 𝑥𝑅𝐶)))
2		exanres3 38806	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑥 ∈ 𝐴 (𝐵 ∈ [𝑥]𝑅 ∧ 𝐶 ∈ [𝑥]𝑅) ↔ ∃𝑥 ∈ 𝐴 (𝑥𝑅𝐵 ∧ 𝑥𝑅𝐶)))
3	1, 2	bitr4d 284	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ↾ 𝐴)𝐶 ↔ ∃𝑥 ∈ 𝐴 (𝐵 ∈ [𝑥]𝑅 ∧ 𝐶 ∈ [𝑥]𝑅)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2144  ∃wrex 3088   class class class wbr 5102   ↾ cres 5651  [cec 8678   ≀ ccoss 38687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-xp 5655  df-cnv 5657  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ec 8682  df-coss 39005

This theorem is referenced by:  relbrcoss  39040