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Theorem cnvtrrel 43661
Description: The converse of a transitive relation is a transitive relation. (Contributed by RP, 25-Dec-2019.)
Assertion
Ref Expression
cnvtrrel ((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)

Proof of Theorem cnvtrrel
StepHypRef Expression
1 cnvss 5881 . . 3 ((𝑆𝑆) ⊆ 𝑆(𝑆𝑆) ⊆ 𝑆)
2 cnvss 5881 . . . 4 ((𝑆𝑆) ⊆ 𝑆(𝑆𝑆) ⊆ 𝑆)
3 cnvco 5894 . . . . . . . . 9 (𝑆𝑆) = (𝑆𝑆)
43cnveqi 5883 . . . . . . . 8 (𝑆𝑆) = (𝑆𝑆)
5 cnvco 5894 . . . . . . . 8 (𝑆𝑆) = (𝑆𝑆)
6 cocnvcnv1 6275 . . . . . . . . 9 (𝑆𝑆) = (𝑆𝑆)
7 cocnvcnv2 6276 . . . . . . . . 9 (𝑆𝑆) = (𝑆𝑆)
86, 7eqtri 2764 . . . . . . . 8 (𝑆𝑆) = (𝑆𝑆)
94, 5, 83eqtri 2768 . . . . . . 7 (𝑆𝑆) = (𝑆𝑆)
109sseq1i 4011 . . . . . 6 ((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)
1110biimpi 216 . . . . 5 ((𝑆𝑆) ⊆ 𝑆 → (𝑆𝑆) ⊆ 𝑆)
12 cnvcnvss 6212 . . . . 5 𝑆𝑆
1311, 12sstrdi 3995 . . . 4 ((𝑆𝑆) ⊆ 𝑆 → (𝑆𝑆) ⊆ 𝑆)
142, 13syl 17 . . 3 ((𝑆𝑆) ⊆ 𝑆 → (𝑆𝑆) ⊆ 𝑆)
151, 14impbii 209 . 2 ((𝑆𝑆) ⊆ 𝑆(𝑆𝑆) ⊆ 𝑆)
163sseq1i 4011 . 2 ((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)
1715, 16bitri 275 1 ((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wss 3950  ccnv 5682  ccom 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5142  df-opab 5204  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695
This theorem is referenced by: (None)
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