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Theorem cnvtrrel 36981
 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 5216 . . 3 ((𝑆𝑆) ⊆ 𝑆(𝑆𝑆) ⊆ 𝑆)
2 cnvss 5216 . . . 4 ((𝑆𝑆) ⊆ 𝑆(𝑆𝑆) ⊆ 𝑆)
3 cnvco 5230 . . . . . . . . 9 (𝑆𝑆) = (𝑆𝑆)
43cnveqi 5219 . . . . . . . 8 (𝑆𝑆) = (𝑆𝑆)
5 cnvco 5230 . . . . . . . 8 (𝑆𝑆) = (𝑆𝑆)
6 cocnvcnv1 5563 . . . . . . . . 9 (𝑆𝑆) = (𝑆𝑆)
7 cocnvcnv2 5564 . . . . . . . . 9 (𝑆𝑆) = (𝑆𝑆)
86, 7eqtri 2632 . . . . . . . 8 (𝑆𝑆) = (𝑆𝑆)
94, 5, 83eqtri 2636 . . . . . . 7 (𝑆𝑆) = (𝑆𝑆)
109sseq1i 3592 . . . . . 6 ((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)
1110biimpi 205 . . . . 5 ((𝑆𝑆) ⊆ 𝑆 → (𝑆𝑆) ⊆ 𝑆)
12 cnvcnvss 5507 . . . . 5 𝑆𝑆
1311, 12syl6ss 3580 . . . 4 ((𝑆𝑆) ⊆ 𝑆 → (𝑆𝑆) ⊆ 𝑆)
142, 13syl 17 . . 3 ((𝑆𝑆) ⊆ 𝑆 → (𝑆𝑆) ⊆ 𝑆)
151, 14impbii 198 . 2 ((𝑆𝑆) ⊆ 𝑆(𝑆𝑆) ⊆ 𝑆)
163sseq1i 3592 . 2 ((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)
1715, 16bitri 263 1 ((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ⊆ wss 3540  ◡ccnv 5037   ∘ ccom 5042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050 This theorem is referenced by: (None)
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