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Theorem relcnvtr 4918
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 4591 . . 3 (𝑅𝑅) = (𝑅𝑅)
2 cnvss 4579 . . 3 ((𝑅𝑅) ⊆ 𝑅(𝑅𝑅) ⊆ 𝑅)
31, 2syl5eqssr 3060 . 2 ((𝑅𝑅) ⊆ 𝑅 → (𝑅𝑅) ⊆ 𝑅)
4 cnvco 4591 . . . 4 (𝑅𝑅) = (𝑅𝑅)
5 cnvss 4579 . . . 4 ((𝑅𝑅) ⊆ 𝑅(𝑅𝑅) ⊆ 𝑅)
6 sseq1 3036 . . . . 5 ((𝑅𝑅) = (𝑅𝑅) → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
7 dfrel2 4849 . . . . . . 7 (Rel 𝑅𝑅 = 𝑅)
8 coeq1 4563 . . . . . . . . . 10 (𝑅 = 𝑅 → (𝑅𝑅) = (𝑅𝑅))
9 coeq2 4564 . . . . . . . . . 10 (𝑅 = 𝑅 → (𝑅𝑅) = (𝑅𝑅))
108, 9eqtrd 2117 . . . . . . . . 9 (𝑅 = 𝑅 → (𝑅𝑅) = (𝑅𝑅))
11 id 19 . . . . . . . . 9 (𝑅 = 𝑅𝑅 = 𝑅)
1210, 11sseq12d 3044 . . . . . . . 8 (𝑅 = 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
1312biimpd 142 . . . . . . 7 (𝑅 = 𝑅 → ((𝑅𝑅) ⊆ 𝑅 → (𝑅𝑅) ⊆ 𝑅))
147, 13sylbi 119 . . . . . 6 (Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 → (𝑅𝑅) ⊆ 𝑅))
1514com12 30 . . . . 5 ((𝑅𝑅) ⊆ 𝑅 → (Rel 𝑅 → (𝑅𝑅) ⊆ 𝑅))
166, 15syl6bi 161 . . . 4 ((𝑅𝑅) = (𝑅𝑅) → ((𝑅𝑅) ⊆ 𝑅 → (Rel 𝑅 → (𝑅𝑅) ⊆ 𝑅)))
174, 5, 16mpsyl 64 . . 3 ((𝑅𝑅) ⊆ 𝑅 → (Rel 𝑅 → (𝑅𝑅) ⊆ 𝑅))
1817com12 30 . 2 (Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 → (𝑅𝑅) ⊆ 𝑅))
193, 18impbid2 141 1 (Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1287  wss 2988  ccnv 4412  ccom 4417  Rel wrel 4418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422
This theorem is referenced by: (None)
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