Theorem cnvtrrel 44860

Description: The converse of a transitive relation is a transitive relation. (Contributed by RP, 25-Dec-2019.)

Assertion

Ref	Expression
cnvtrrel	⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆)

Proof of Theorem cnvtrrel

Step	Hyp	Ref	Expression
1		cnvss 5856	. . 3 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 → ◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆)
2		cnvss 5856	. . . 4 ⊢ (◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆 → ◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆)
3		cnvco 5873	. . . . . . . . 9 ⊢ ◡(𝑆 ∘ 𝑆) = (◡𝑆 ∘ ◡𝑆)
4	3	cnveqi 5858	. . . . . . . 8 ⊢ ◡◡(𝑆 ∘ 𝑆) = ◡(◡𝑆 ∘ ◡𝑆)
5		cnvco 5873	. . . . . . . 8 ⊢ ◡(◡𝑆 ∘ ◡𝑆) = (◡◡𝑆 ∘ ◡◡𝑆)
6		cocnvcnv1 6271	. . . . . . . . 9 ⊢ (◡◡𝑆 ∘ ◡◡𝑆) = (𝑆 ∘ ◡◡𝑆)
7		cocnvcnv2 6272	. . . . . . . . 9 ⊢ (𝑆 ∘ ◡◡𝑆) = (𝑆 ∘ 𝑆)
8	6, 7	eqtri 2786	. . . . . . . 8 ⊢ (◡◡𝑆 ∘ ◡◡𝑆) = (𝑆 ∘ 𝑆)
9	4, 5, 8	3eqtri 2790	. . . . . . 7 ⊢ ◡◡(𝑆 ∘ 𝑆) = (𝑆 ∘ 𝑆)
10	9	sseq1i 3941	. . . . . 6 ⊢ (◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆 ↔ (𝑆 ∘ 𝑆) ⊆ ◡◡𝑆)
11	10	biimpi 219	. . . . 5 ⊢ (◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆 → (𝑆 ∘ 𝑆) ⊆ ◡◡𝑆)
12		cnvcnvss 6197	. . . . 5 ⊢ ◡◡𝑆 ⊆ 𝑆
13	11, 12	sstrdi 3925	. . . 4 ⊢ (◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
14	2, 13	syl 18	. . 3 ⊢ (◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
15	1, 14	impbii 212	. 2 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ ◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆)
16	3	sseq1i 3941	. 2 ⊢ (◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆)
17	15, 16	bitri 278	1 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆)

This proof depends on syntax axioms:   ↔ wb 209   ⊆ wss 3900  ^◡ccnv 5654   ∘ ccom 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667

This theorem is used by: (None)