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Theorem cnvtrrel 39893
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 5736 . . 3 ((𝑆𝑆) ⊆ 𝑆(𝑆𝑆) ⊆ 𝑆)
2 cnvss 5736 . . . 4 ((𝑆𝑆) ⊆ 𝑆(𝑆𝑆) ⊆ 𝑆)
3 cnvco 5749 . . . . . . . . 9 (𝑆𝑆) = (𝑆𝑆)
43cnveqi 5738 . . . . . . . 8 (𝑆𝑆) = (𝑆𝑆)
5 cnvco 5749 . . . . . . . 8 (𝑆𝑆) = (𝑆𝑆)
6 cocnvcnv1 6103 . . . . . . . . 9 (𝑆𝑆) = (𝑆𝑆)
7 cocnvcnv2 6104 . . . . . . . . 9 (𝑆𝑆) = (𝑆𝑆)
86, 7eqtri 2841 . . . . . . . 8 (𝑆𝑆) = (𝑆𝑆)
94, 5, 83eqtri 2845 . . . . . . 7 (𝑆𝑆) = (𝑆𝑆)
109sseq1i 3992 . . . . . 6 ((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)
1110biimpi 217 . . . . 5 ((𝑆𝑆) ⊆ 𝑆 → (𝑆𝑆) ⊆ 𝑆)
12 cnvcnvss 6044 . . . . 5 𝑆𝑆
1311, 12sstrdi 3976 . . . 4 ((𝑆𝑆) ⊆ 𝑆 → (𝑆𝑆) ⊆ 𝑆)
142, 13syl 17 . . 3 ((𝑆𝑆) ⊆ 𝑆 → (𝑆𝑆) ⊆ 𝑆)
151, 14impbii 210 . 2 ((𝑆𝑆) ⊆ 𝑆(𝑆𝑆) ⊆ 𝑆)
163sseq1i 3992 . 2 ((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)
1715, 16bitri 276 1 ((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wss 3933  ccnv 5547  ccom 5552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560
This theorem is referenced by: (None)
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