Theorem relbrcoss 38903

Description: 𝐴 and 𝐵 are cosets by relation 𝑅: a binary relation. (Contributed by Peter Mazsa, 22-Apr-2021.)

Assertion

Ref	Expression
relbrcoss	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (Rel 𝑅 → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅))))

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

Proof of Theorem relbrcoss

Step	Hyp	Ref	Expression
1		resdm 5978	. . . . . 6 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
2	1	cosseqd 38885	. . . . 5 ⊢ (Rel 𝑅 → ≀ (𝑅 ↾ dom 𝑅) = ≀ 𝑅)
3	2	breqd 5083	. . . 4 ⊢ (Rel 𝑅 → (𝐴 ≀ (𝑅 ↾ dom 𝑅)𝐵 ↔ 𝐴 ≀ 𝑅𝐵))
4	3	adantl 482	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Rel 𝑅) → (𝐴 ≀ (𝑅 ↾ dom 𝑅)𝐵 ↔ 𝐴 ≀ 𝑅𝐵))
5		br1cossres2 38897	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ (𝑅 ↾ dom 𝑅)𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅)))
6	5	adantr 481	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Rel 𝑅) → (𝐴 ≀ (𝑅 ↾ dom 𝑅)𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅)))
7	4, 6	bitr3d 282	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Rel 𝑅) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅)))
8	7	ex 413	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (Rel 𝑅 → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∈ wcel 2119  ∃wrex 3063   class class class wbr 5072  dom cdm 5618   ↾ cres 5620  Rel wrel 5623  [cec 8631   ≀ ccoss 38550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ec 8635  df-coss 38868

This theorem is referenced by:  disjlem18  39270