Theorem relcnvtr 6087

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 1101	. 2 ⊢ ((Rel 𝑅 ∧ Rel 𝑅 ∧ Rel 𝑅) ↔ Rel 𝑅)
2		relcnvtrg 6086	. 2 ⊢ ((Rel 𝑅 ∧ Rel 𝑅 ∧ Rel 𝑅) → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅))
3	1, 2	sylbir 238	1 ⊢ (Rel 𝑅 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1084   ⊆ wss 3881  ^◡ccnv 5518   ∘ ccom 5523  Rel wrel 5524

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528

This theorem is referenced by: (None)