Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cnvtrrel Structured version   Visualization version   GIF version

Theorem cnvtrrel 40150
 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 5719 . . 3 ((𝑆𝑆) ⊆ 𝑆(𝑆𝑆) ⊆ 𝑆)
2 cnvss 5719 . . . 4 ((𝑆𝑆) ⊆ 𝑆(𝑆𝑆) ⊆ 𝑆)
3 cnvco 5732 . . . . . . . . 9 (𝑆𝑆) = (𝑆𝑆)
43cnveqi 5721 . . . . . . . 8 (𝑆𝑆) = (𝑆𝑆)
5 cnvco 5732 . . . . . . . 8 (𝑆𝑆) = (𝑆𝑆)
6 cocnvcnv1 6086 . . . . . . . . 9 (𝑆𝑆) = (𝑆𝑆)
7 cocnvcnv2 6087 . . . . . . . . 9 (𝑆𝑆) = (𝑆𝑆)
86, 7eqtri 2843 . . . . . . . 8 (𝑆𝑆) = (𝑆𝑆)
94, 5, 83eqtri 2847 . . . . . . 7 (𝑆𝑆) = (𝑆𝑆)
109sseq1i 3974 . . . . . 6 ((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)
1110biimpi 218 . . . . 5 ((𝑆𝑆) ⊆ 𝑆 → (𝑆𝑆) ⊆ 𝑆)
12 cnvcnvss 6027 . . . . 5 𝑆𝑆
1311, 12sstrdi 3958 . . . 4 ((𝑆𝑆) ⊆ 𝑆 → (𝑆𝑆) ⊆ 𝑆)
142, 13syl 17 . . 3 ((𝑆𝑆) ⊆ 𝑆 → (𝑆𝑆) ⊆ 𝑆)
151, 14impbii 211 . 2 ((𝑆𝑆) ⊆ 𝑆(𝑆𝑆) ⊆ 𝑆)
163sseq1i 3974 . 2 ((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)
1715, 16bitri 277 1 ((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ⊆ wss 3913  ◡ccnv 5530   ∘ ccom 5535 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-br 5043  df-opab 5105  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator