Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  releleccnv Structured version   Visualization version   GIF version

Theorem releleccnv 34264
 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 5613 . . 3 Rel 𝑅
2 relelec 7905 . . 3 (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
31, 2ax-mp 5 . 2 (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴)
4 relbrcnvg 5614 . 2 (Rel 𝑅 → (𝐵𝑅𝐴𝐴𝑅𝐵))
53, 4syl5bb 272 1 (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅𝐴𝑅𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 2103   class class class wbr 4760  ◡ccnv 5217  Rel wrel 5223  [cec 7860 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-xp 5224  df-rel 5225  df-cnv 5226  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-ec 7864 This theorem is referenced by:  releccnveq  34265
 Copyright terms: Public domain W3C validator