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Theorem relcnvtr 4865
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 4545 . . 3 (𝑅𝑅) = (𝑅𝑅)
2 cnvss 4533 . . 3 ((𝑅𝑅) ⊆ 𝑅(𝑅𝑅) ⊆ 𝑅)
31, 2syl5eqssr 3015 . 2 ((𝑅𝑅) ⊆ 𝑅 → (𝑅𝑅) ⊆ 𝑅)
4 cnvco 4545 . . . 4 (𝑅𝑅) = (𝑅𝑅)
5 cnvss 4533 . . . 4 ((𝑅𝑅) ⊆ 𝑅(𝑅𝑅) ⊆ 𝑅)
6 sseq1 2991 . . . . 5 ((𝑅𝑅) = (𝑅𝑅) → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
7 dfrel2 4796 . . . . . . 7 (Rel 𝑅𝑅 = 𝑅)
8 coeq1 4518 . . . . . . . . . 10 (𝑅 = 𝑅 → (𝑅𝑅) = (𝑅𝑅))
9 coeq2 4519 . . . . . . . . . 10 (𝑅 = 𝑅 → (𝑅𝑅) = (𝑅𝑅))
108, 9eqtrd 2086 . . . . . . . . 9 (𝑅 = 𝑅 → (𝑅𝑅) = (𝑅𝑅))
11 id 19 . . . . . . . . 9 (𝑅 = 𝑅𝑅 = 𝑅)
1210, 11sseq12d 2999 . . . . . . . 8 (𝑅 = 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
1312biimpd 136 . . . . . . 7 (𝑅 = 𝑅 → ((𝑅𝑅) ⊆ 𝑅 → (𝑅𝑅) ⊆ 𝑅))
147, 13sylbi 118 . . . . . 6 (Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 → (𝑅𝑅) ⊆ 𝑅))
1514com12 30 . . . . 5 ((𝑅𝑅) ⊆ 𝑅 → (Rel 𝑅 → (𝑅𝑅) ⊆ 𝑅))
166, 15syl6bi 156 . . . 4 ((𝑅𝑅) = (𝑅𝑅) → ((𝑅𝑅) ⊆ 𝑅 → (Rel 𝑅 → (𝑅𝑅) ⊆ 𝑅)))
174, 5, 16mpsyl 63 . . 3 ((𝑅𝑅) ⊆ 𝑅 → (Rel 𝑅 → (𝑅𝑅) ⊆ 𝑅))
1817com12 30 . 2 (Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 → (𝑅𝑅) ⊆ 𝑅))
193, 18impbid2 135 1 (Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1257  wss 2942  ccnv 4369  ccom 4374  Rel wrel 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3790  df-opab 3844  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379
This theorem is referenced by: (None)
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