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Theorem brcosscnv 37984
Description: 𝐴 and 𝐵 are cosets by converse 𝑅: a binary relation. (Contributed by Peter Mazsa, 23-Jan-2019.)
Assertion
Ref Expression
brcosscnv ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑉   𝑥,𝑊

Proof of Theorem brcosscnv
StepHypRef Expression
1 brcoss 37943 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑥(𝑥𝑅𝐴𝑥𝑅𝐵)))
2 brcnvg 5886 . . . . 5 ((𝑥 ∈ V ∧ 𝐴𝑉) → (𝑥𝑅𝐴𝐴𝑅𝑥))
32el2v1 37731 . . . 4 (𝐴𝑉 → (𝑥𝑅𝐴𝐴𝑅𝑥))
4 brcnvg 5886 . . . . 5 ((𝑥 ∈ V ∧ 𝐵𝑊) → (𝑥𝑅𝐵𝐵𝑅𝑥))
54el2v1 37731 . . . 4 (𝐵𝑊 → (𝑥𝑅𝐵𝐵𝑅𝑥))
63, 5bi2anan9 636 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝑥𝑅𝐴𝑥𝑅𝐵) ↔ (𝐴𝑅𝑥𝐵𝑅𝑥)))
76exbidv 1916 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑥(𝑥𝑅𝐴𝑥𝑅𝐵) ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
81, 7bitrd 278 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wex 1773  wcel 2098  Vcvv 3473   class class class wbr 5152  ccnv 5681  ccoss 37689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-cnv 5690  df-coss 37923
This theorem is referenced by:  brcosscnv2  37985  br1cosscnvxrn  37986
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