Theorem relbrcoss 38709

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 5985	. . . . . 6 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
2	1	cosseqd 38691	. . . . 5 ⊢ (Rel 𝑅 → ≀ (𝑅 ↾ dom 𝑅) = ≀ 𝑅)
3	2	breqd 5109	. . . 4 ⊢ (Rel 𝑅 → (𝐴 ≀ (𝑅 ↾ dom 𝑅)𝐵 ↔ 𝐴 ≀ 𝑅𝐵))
4	3	adantl 481	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Rel 𝑅) → (𝐴 ≀ (𝑅 ↾ dom 𝑅)𝐵 ↔ 𝐴 ≀ 𝑅𝐵))
5		br1cossres2 38703	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ (𝑅 ↾ dom 𝑅)𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅)))
6	5	adantr 480	. . 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 281	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Rel 𝑅) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅)))
8	7	ex 412	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (Rel 𝑅 → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2113  ∃wrex 3060   class class class wbr 5098  dom cdm 5624   ↾ cres 5626  Rel wrel 5629  [cec 8633   ≀ ccoss 38383

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 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8637  df-coss 38674

This theorem is referenced by:  disjlem18  39059