Theorem relcnvtr 6251

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 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅))

Proof of Theorem relcnvtr

Step	Hyp	Ref	Expression
1		3anidm 1115	. 2 ⊢ ((Rel 𝑅 ∧ Rel 𝑅 ∧ Rel 𝑅) ↔ Rel 𝑅)
2		relcnvtrg 6250	. 2 ⊢ ((Rel 𝑅 ∧ Rel 𝑅 ∧ Rel 𝑅) → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅))
3	1, 2	sylbir 237	1 ⊢ (Rel 𝑅 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1097   ⊆ wss 3904  ^◡ccnv 5644   ∘ ccom 5649  Rel wrel 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654

This theorem is referenced by: (None)