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Theorem relcnvtr 6205
Description: A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Peter Mazsa, 17-Oct-2023.)
Assertion
Ref Expression
relcnvtr (Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))

Proof of Theorem relcnvtr
StepHypRef Expression
1 3anidm 1103 . 2 ((Rel 𝑅 ∧ Rel 𝑅 ∧ Rel 𝑅) ↔ Rel 𝑅)
2 relcnvtrg 6204 . 2 ((Rel 𝑅 ∧ Rel 𝑅 ∧ Rel 𝑅) → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
31, 2sylbir 234 1 (Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1086  wss 3898  ccnv 5619  ccom 5624  Rel wrel 5625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629
This theorem is referenced by: (None)
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