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Theorem releleccnv 35626
Description: Elementhood in a converse 𝑅-coset when 𝑅 is a relation. (Contributed by Peter Mazsa, 9-Dec-2018.)
Assertion
Ref Expression
releleccnv (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅𝐴𝑅𝐵))

Proof of Theorem releleccnv
StepHypRef Expression
1 relcnv 5954 . . 3 Rel 𝑅
2 relelec 8330 . . 3 (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
31, 2ax-mp 5 . 2 (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴)
4 relbrcnvg 5955 . 2 (Rel 𝑅 → (𝐵𝑅𝐴𝐴𝑅𝐵))
53, 4syl5bb 286 1 (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅𝐴𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2115   class class class wbr 5052  ccnv 5541  Rel wrel 5547  [cec 8283
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ec 8287
This theorem is referenced by:  releccnveq  35627
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