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Theorem brcosscnvcoss 34496
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 1947 . . 3 (∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴))
21a1i 11 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴)))
3 brcoss 34493 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
4 brcoss 34493 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐵𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴)))
54ancoms 448 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵𝑢𝑅𝐴)))
62, 3, 53bitr4d 302 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wex 1859  wcel 2155   class class class wbr 4837  ccoss 34287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pr 5090
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-rab 3101  df-v 3389  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-sn 4365  df-pr 4367  df-op 4371  df-br 4838  df-opab 4900  df-coss 34476
This theorem is referenced by:  cocossss  34498  cnvcosseq  34499  rncossdmcoss  34512  symrelcoss3  34522  eleccossin  34540
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