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Theorem cnvtrrel 38279
Description: The converse of a transitive relation is a transitive relation. (Contributed by Richard Penner, 25-Dec-2019.)
Assertion
Ref Expression
cnvtrrel ((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)

Proof of Theorem cnvtrrel
StepHypRef Expression
1 cnvss 5327 . . 3 ((𝑆𝑆) ⊆ 𝑆(𝑆𝑆) ⊆ 𝑆)
2 cnvss 5327 . . . 4 ((𝑆𝑆) ⊆ 𝑆(𝑆𝑆) ⊆ 𝑆)
3 cnvco 5340 . . . . . . . . 9 (𝑆𝑆) = (𝑆𝑆)
43cnveqi 5329 . . . . . . . 8 (𝑆𝑆) = (𝑆𝑆)
5 cnvco 5340 . . . . . . . 8 (𝑆𝑆) = (𝑆𝑆)
6 cocnvcnv1 5684 . . . . . . . . 9 (𝑆𝑆) = (𝑆𝑆)
7 cocnvcnv2 5685 . . . . . . . . 9 (𝑆𝑆) = (𝑆𝑆)
86, 7eqtri 2673 . . . . . . . 8 (𝑆𝑆) = (𝑆𝑆)
94, 5, 83eqtri 2677 . . . . . . 7 (𝑆𝑆) = (𝑆𝑆)
109sseq1i 3662 . . . . . 6 ((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)
1110biimpi 206 . . . . 5 ((𝑆𝑆) ⊆ 𝑆 → (𝑆𝑆) ⊆ 𝑆)
12 cnvcnvss 5624 . . . . 5 𝑆𝑆
1311, 12syl6ss 3648 . . . 4 ((𝑆𝑆) ⊆ 𝑆 → (𝑆𝑆) ⊆ 𝑆)
142, 13syl 17 . . 3 ((𝑆𝑆) ⊆ 𝑆 → (𝑆𝑆) ⊆ 𝑆)
151, 14impbii 199 . 2 ((𝑆𝑆) ⊆ 𝑆(𝑆𝑆) ⊆ 𝑆)
163sseq1i 3662 . 2 ((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)
1715, 16bitri 264 1 ((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wss 3607  ccnv 5142  ccom 5147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155
This theorem is referenced by: (None)
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