Theorem cnvtrrel 40371

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 5707	. . 3 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 → ◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆)
2		cnvss 5707	. . . 4 ⊢ (◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆 → ◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆)
3		cnvco 5720	. . . . . . . . 9 ⊢ ◡(𝑆 ∘ 𝑆) = (◡𝑆 ∘ ◡𝑆)
4	3	cnveqi 5709	. . . . . . . 8 ⊢ ◡◡(𝑆 ∘ 𝑆) = ◡(◡𝑆 ∘ ◡𝑆)
5		cnvco 5720	. . . . . . . 8 ⊢ ◡(◡𝑆 ∘ ◡𝑆) = (◡◡𝑆 ∘ ◡◡𝑆)
6		cocnvcnv1 6077	. . . . . . . . 9 ⊢ (◡◡𝑆 ∘ ◡◡𝑆) = (𝑆 ∘ ◡◡𝑆)
7		cocnvcnv2 6078	. . . . . . . . 9 ⊢ (𝑆 ∘ ◡◡𝑆) = (𝑆 ∘ 𝑆)
8	6, 7	eqtri 2821	. . . . . . . 8 ⊢ (◡◡𝑆 ∘ ◡◡𝑆) = (𝑆 ∘ 𝑆)
9	4, 5, 8	3eqtri 2825	. . . . . . 7 ⊢ ◡◡(𝑆 ∘ 𝑆) = (𝑆 ∘ 𝑆)
10	9	sseq1i 3943	. . . . . 6 ⊢ (◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆 ↔ (𝑆 ∘ 𝑆) ⊆ ◡◡𝑆)
11	10	biimpi 219	. . . . 5 ⊢ (◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆 → (𝑆 ∘ 𝑆) ⊆ ◡◡𝑆)
12		cnvcnvss 6018	. . . . 5 ⊢ ◡◡𝑆 ⊆ 𝑆
13	11, 12	sstrdi 3927	. . . 4 ⊢ (◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
14	2, 13	syl 17	. . 3 ⊢ (◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
15	1, 14	impbii 212	. 2 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ ◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆)
16	3	sseq1i 3943	. 2 ⊢ (◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆)
17	15, 16	bitri 278	1 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆)

Syntax hints:   ↔ wb 209   ⊆ wss 3881  ^◡ccnv 5518   ∘ ccom 5523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531

This theorem is referenced by: (None)