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Theorem relbrcoss 35566
Description: 𝐴 and 𝐵 are cosets by relation 𝑅: a binary relation. (Contributed by Peter Mazsa, 22-Apr-2021.)
Assertion
Ref Expression
relbrcoss ((𝐴𝑉𝐵𝑊) → (Rel 𝑅 → (𝐴𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑉   𝑥,𝑊

Proof of Theorem relbrcoss
StepHypRef Expression
1 resdm 5890 . . . . . 6 (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
21cosseqd 35553 . . . . 5 (Rel 𝑅 → ≀ (𝑅 ↾ dom 𝑅) = ≀ 𝑅)
32breqd 5068 . . . 4 (Rel 𝑅 → (𝐴 ≀ (𝑅 ↾ dom 𝑅)𝐵𝐴𝑅𝐵))
43adantl 482 . . 3 (((𝐴𝑉𝐵𝑊) ∧ Rel 𝑅) → (𝐴 ≀ (𝑅 ↾ dom 𝑅)𝐵𝐴𝑅𝐵))
5 br1cossres2 35565 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 ≀ (𝑅 ↾ dom 𝑅)𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅)))
65adantr 481 . . 3 (((𝐴𝑉𝐵𝑊) ∧ Rel 𝑅) → (𝐴 ≀ (𝑅 ↾ dom 𝑅)𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅)))
74, 6bitr3d 282 . 2 (((𝐴𝑉𝐵𝑊) ∧ Rel 𝑅) → (𝐴𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅)))
87ex 413 1 ((𝐴𝑉𝐵𝑊) → (Rel 𝑅 → (𝐴𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  wrex 3136   class class class wbr 5057  dom cdm 5548  cres 5550  Rel wrel 5553  [cec 8276  ccoss 35334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ec 8280  df-coss 35539
This theorem is referenced by: (None)
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