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Theorem relcnvtr 4783
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 4463 . . 3 (𝑅𝑅) = (𝑅𝑅)
2 cnvss 4451 . . 3 ((𝑅𝑅) ⊆ 𝑅(𝑅𝑅) ⊆ 𝑅)
31, 2syl5eqssr 2984 . 2 ((𝑅𝑅) ⊆ 𝑅 → (𝑅𝑅) ⊆ 𝑅)
4 cnvco 4463 . . . 4 (𝑅𝑅) = (𝑅𝑅)
5 cnvss 4451 . . . 4 ((𝑅𝑅) ⊆ 𝑅(𝑅𝑅) ⊆ 𝑅)
6 sseq1 2960 . . . . 5 ((𝑅𝑅) = (𝑅𝑅) → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
7 dfrel2 4714 . . . . . . 7 (Rel 𝑅𝑅 = 𝑅)
8 coeq1 4436 . . . . . . . . . 10 (𝑅 = 𝑅 → (𝑅𝑅) = (𝑅𝑅))
9 coeq2 4437 . . . . . . . . . 10 (𝑅 = 𝑅 → (𝑅𝑅) = (𝑅𝑅))
108, 9eqtrd 2069 . . . . . . . . 9 (𝑅 = 𝑅 → (𝑅𝑅) = (𝑅𝑅))
11 id 19 . . . . . . . . 9 (𝑅 = 𝑅𝑅 = 𝑅)
1210, 11sseq12d 2968 . . . . . . . 8 (𝑅 = 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
1312biimpd 132 . . . . . . 7 (𝑅 = 𝑅 → ((𝑅𝑅) ⊆ 𝑅 → (𝑅𝑅) ⊆ 𝑅))
147, 13sylbi 114 . . . . . 6 (Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 → (𝑅𝑅) ⊆ 𝑅))
1514com12 27 . . . . 5 ((𝑅𝑅) ⊆ 𝑅 → (Rel 𝑅 → (𝑅𝑅) ⊆ 𝑅))
166, 15syl6bi 152 . . . 4 ((𝑅𝑅) = (𝑅𝑅) → ((𝑅𝑅) ⊆ 𝑅 → (Rel 𝑅 → (𝑅𝑅) ⊆ 𝑅)))
174, 5, 16mpsyl 59 . . 3 ((𝑅𝑅) ⊆ 𝑅 → (Rel 𝑅 → (𝑅𝑅) ⊆ 𝑅))
1817com12 27 . 2 (Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 → (𝑅𝑅) ⊆ 𝑅))
193, 18impbid2 131 1 (Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wss 2911  ccnv 4287  ccom 4292  Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297
This theorem is referenced by: (None)
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