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Theorem brcosscnvcoss 35797
Description: For sets, the 𝐴 and 𝐵 cosets by 𝑅 binary relation and the 𝐵 and 𝐴 cosets by 𝑅 binary relation are the same. (Contributed by Peter Mazsa, 27-Dec-2018.)
Assertion
Ref Expression
brcosscnvcoss ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝐵𝑅𝐴))

Proof of Theorem brcosscnvcoss
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 exancom 1862 . . 3 (∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴))
21a1i 11 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴)))
3 brcoss 35794 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
4 brcoss 35794 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐵𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴)))
54ancoms 462 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴)))
62, 3, 53bitr4d 314 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1781  wcel 2114   class class class wbr 5042  ccoss 35571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-coss 35777
This theorem is referenced by:  cocossss  35799  cnvcosseq  35800  rncossdmcoss  35813  symrelcoss3  35823  eleccossin  35841
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