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Theorem relcnvtr 5693
Description: A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
relcnvtr (Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))

Proof of Theorem relcnvtr
StepHypRef Expression
1 cnvco 5340 . . 3 (𝑅𝑅) = (𝑅𝑅)
2 cnvss 5327 . . 3 ((𝑅𝑅) ⊆ 𝑅(𝑅𝑅) ⊆ 𝑅)
31, 2syl5eqssr 3683 . 2 ((𝑅𝑅) ⊆ 𝑅 → (𝑅𝑅) ⊆ 𝑅)
4 cnvco 5340 . . . 4 (𝑅𝑅) = (𝑅𝑅)
5 cnvss 5327 . . . 4 ((𝑅𝑅) ⊆ 𝑅(𝑅𝑅) ⊆ 𝑅)
6 sseq1 3659 . . . . 5 ((𝑅𝑅) = (𝑅𝑅) → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
7 dfrel2 5618 . . . . . . 7 (Rel 𝑅𝑅 = 𝑅)
8 coeq1 5312 . . . . . . . . . 10 (𝑅 = 𝑅 → (𝑅𝑅) = (𝑅𝑅))
9 coeq2 5313 . . . . . . . . . 10 (𝑅 = 𝑅 → (𝑅𝑅) = (𝑅𝑅))
108, 9eqtrd 2685 . . . . . . . . 9 (𝑅 = 𝑅 → (𝑅𝑅) = (𝑅𝑅))
11 id 22 . . . . . . . . 9 (𝑅 = 𝑅𝑅 = 𝑅)
1210, 11sseq12d 3667 . . . . . . . 8 (𝑅 = 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
1312biimpd 219 . . . . . . 7 (𝑅 = 𝑅 → ((𝑅𝑅) ⊆ 𝑅 → (𝑅𝑅) ⊆ 𝑅))
147, 13sylbi 207 . . . . . 6 (Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 → (𝑅𝑅) ⊆ 𝑅))
1514com12 32 . . . . 5 ((𝑅𝑅) ⊆ 𝑅 → (Rel 𝑅 → (𝑅𝑅) ⊆ 𝑅))
166, 15syl6bi 243 . . . 4 ((𝑅𝑅) = (𝑅𝑅) → ((𝑅𝑅) ⊆ 𝑅 → (Rel 𝑅 → (𝑅𝑅) ⊆ 𝑅)))
174, 5, 16mpsyl 68 . . 3 ((𝑅𝑅) ⊆ 𝑅 → (Rel 𝑅 → (𝑅𝑅) ⊆ 𝑅))
1817com12 32 . 2 (Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 → (𝑅𝑅) ⊆ 𝑅))
193, 18impbid2 216 1 (Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wss 3607  ccnv 5142  ccom 5147  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152
This theorem is referenced by: (None)
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