Theorem cnvtrrel 43660

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 5863	. . 3 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 → ◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆)
2		cnvss 5863	. . . 4 ⊢ (◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆 → ◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆)
3		cnvco 5876	. . . . . . . . 9 ⊢ ◡(𝑆 ∘ 𝑆) = (◡𝑆 ∘ ◡𝑆)
4	3	cnveqi 5865	. . . . . . . 8 ⊢ ◡◡(𝑆 ∘ 𝑆) = ◡(◡𝑆 ∘ ◡𝑆)
5		cnvco 5876	. . . . . . . 8 ⊢ ◡(◡𝑆 ∘ ◡𝑆) = (◡◡𝑆 ∘ ◡◡𝑆)
6		cocnvcnv1 6257	. . . . . . . . 9 ⊢ (◡◡𝑆 ∘ ◡◡𝑆) = (𝑆 ∘ ◡◡𝑆)
7		cocnvcnv2 6258	. . . . . . . . 9 ⊢ (𝑆 ∘ ◡◡𝑆) = (𝑆 ∘ 𝑆)
8	6, 7	eqtri 2757	. . . . . . . 8 ⊢ (◡◡𝑆 ∘ ◡◡𝑆) = (𝑆 ∘ 𝑆)
9	4, 5, 8	3eqtri 2761	. . . . . . 7 ⊢ ◡◡(𝑆 ∘ 𝑆) = (𝑆 ∘ 𝑆)
10	9	sseq1i 3992	. . . . . 6 ⊢ (◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆 ↔ (𝑆 ∘ 𝑆) ⊆ ◡◡𝑆)
11	10	biimpi 216	. . . . 5 ⊢ (◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆 → (𝑆 ∘ 𝑆) ⊆ ◡◡𝑆)
12		cnvcnvss 6194	. . . . 5 ⊢ ◡◡𝑆 ⊆ 𝑆
13	11, 12	sstrdi 3976	. . . 4 ⊢ (◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
14	2, 13	syl 17	. . 3 ⊢ (◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
15	1, 14	impbii 209	. 2 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ ◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆)
16	3	sseq1i 3992	. 2 ⊢ (◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆)
17	15, 16	bitri 275	1 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆)

Syntax hints:   ↔ wb 206   ⊆ wss 3931  ^◡ccnv 5664   ∘ ccom 5669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677

This theorem is referenced by: (None)