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Theorem relbrcoss 34689
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 5654 . . . . . 6 (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
21cosseqd 34676 . . . . 5 (Rel 𝑅 → ≀ (𝑅 ↾ dom 𝑅) = ≀ 𝑅)
32breqd 4855 . . . 4 (Rel 𝑅 → (𝐴 ≀ (𝑅 ↾ dom 𝑅)𝐵𝐴𝑅𝐵))
43adantl 474 . . 3 (((𝐴𝑉𝐵𝑊) ∧ Rel 𝑅) → (𝐴 ≀ (𝑅 ↾ dom 𝑅)𝐵𝐴𝑅𝐵))
5 br1cossres2 34688 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 ≀ (𝑅 ↾ dom 𝑅)𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅)))
65adantr 473 . . 3 (((𝐴𝑉𝐵𝑊) ∧ Rel 𝑅) → (𝐴 ≀ (𝑅 ↾ dom 𝑅)𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅)))
74, 6bitr3d 273 . 2 (((𝐴𝑉𝐵𝑊) ∧ Rel 𝑅) → (𝐴𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅)))
87ex 402 1 ((𝐴𝑉𝐵𝑊) → (Rel 𝑅 → (𝐴𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wcel 2157  wrex 3091   class class class wbr 4844  dom cdm 5313  cres 5315  Rel wrel 5318  [cec 7981  ccoss 34468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-br 4845  df-opab 4907  df-xp 5319  df-rel 5320  df-cnv 5321  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-ec 7985  df-coss 34662
This theorem is referenced by: (None)
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