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Theorem relcnvtr 4890
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 4568 . . 3 (𝑅𝑅) = (𝑅𝑅)
2 cnvss 4556 . . 3 ((𝑅𝑅) ⊆ 𝑅(𝑅𝑅) ⊆ 𝑅)
31, 2syl5eqssr 3053 . 2 ((𝑅𝑅) ⊆ 𝑅 → (𝑅𝑅) ⊆ 𝑅)
4 cnvco 4568 . . . 4 (𝑅𝑅) = (𝑅𝑅)
5 cnvss 4556 . . . 4 ((𝑅𝑅) ⊆ 𝑅(𝑅𝑅) ⊆ 𝑅)
6 sseq1 3029 . . . . 5 ((𝑅𝑅) = (𝑅𝑅) → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
7 dfrel2 4821 . . . . . . 7 (Rel 𝑅𝑅 = 𝑅)
8 coeq1 4541 . . . . . . . . . 10 (𝑅 = 𝑅 → (𝑅𝑅) = (𝑅𝑅))
9 coeq2 4542 . . . . . . . . . 10 (𝑅 = 𝑅 → (𝑅𝑅) = (𝑅𝑅))
108, 9eqtrd 2115 . . . . . . . . 9 (𝑅 = 𝑅 → (𝑅𝑅) = (𝑅𝑅))
11 id 19 . . . . . . . . 9 (𝑅 = 𝑅𝑅 = 𝑅)
1210, 11sseq12d 3037 . . . . . . . 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 1285  wss 2982  ccnv 4390  ccom 4395  Rel wrel 4396
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400
This theorem is referenced by: (None)
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