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Theorem brcosscnv 38454
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 38413 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑥(𝑥𝑅𝐴𝑥𝑅𝐵)))
2 brcnvg 5893 . . . . 5 ((𝑥 ∈ V ∧ 𝐴𝑉) → (𝑥𝑅𝐴𝐴𝑅𝑥))
32el2v1 38204 . . . 4 (𝐴𝑉 → (𝑥𝑅𝐴𝐴𝑅𝑥))
4 brcnvg 5893 . . . . 5 ((𝑥 ∈ V ∧ 𝐵𝑊) → (𝑥𝑅𝐵𝐵𝑅𝑥))
54el2v1 38204 . . . 4 (𝐵𝑊 → (𝑥𝑅𝐵𝐵𝑅𝑥))
63, 5bi2anan9 638 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝑥𝑅𝐴𝑥𝑅𝐵) ↔ (𝐴𝑅𝑥𝐵𝑅𝑥)))
76exbidv 1919 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑥(𝑥𝑅𝐴𝑥𝑅𝐵) ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
81, 7bitrd 279 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wex 1776  wcel 2106  Vcvv 3478   class class class wbr 5148  ccnv 5688  ccoss 38162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-coss 38393
This theorem is referenced by:  brcosscnv2  38455  br1cosscnvxrn  38456
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