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Theorem brcosscnv 38473
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 38432 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑥(𝑥𝑅𝐴𝑥𝑅𝐵)))
2 brcnvg 5890 . . . . 5 ((𝑥 ∈ V ∧ 𝐴𝑉) → (𝑥𝑅𝐴𝐴𝑅𝑥))
32el2v1 38224 . . . 4 (𝐴𝑉 → (𝑥𝑅𝐴𝐴𝑅𝑥))
4 brcnvg 5890 . . . . 5 ((𝑥 ∈ V ∧ 𝐵𝑊) → (𝑥𝑅𝐵𝐵𝑅𝑥))
54el2v1 38224 . . . 4 (𝐵𝑊 → (𝑥𝑅𝐵𝐵𝑅𝑥))
63, 5bi2anan9 638 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝑥𝑅𝐴𝑥𝑅𝐵) ↔ (𝐴𝑅𝑥𝐵𝑅𝑥)))
76exbidv 1921 . 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 1779  wcel 2108  Vcvv 3480   class class class wbr 5143  ccnv 5684  ccoss 38182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-coss 38412
This theorem is referenced by:  brcosscnv2  38474  br1cosscnvxrn  38475
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