cnvtrrel - Mathbox for Richard Penner

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

Theorem cnvtrrel 38279

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

**Assertion**
Ref	Expression
cnvtrrel	⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ (^◡𝑆 ∘ ^◡𝑆) ⊆ ^◡𝑆)

**Proof of Theorem cnvtrrel**
Step	Hyp	Ref	Expression
1		cnvss 5327	. . 3 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 → ^◡(𝑆 ∘ 𝑆) ⊆ ^◡𝑆)
2		cnvss 5327	. . . 4 ⊢ (^◡(𝑆 ∘ 𝑆) ⊆ ^◡𝑆 → ^◡^◡(𝑆 ∘ 𝑆) ⊆ ^◡^◡𝑆)
3		cnvco 5340	. . . . . . . . 9 ⊢ ^◡(𝑆 ∘ 𝑆) = (^◡𝑆 ∘ ^◡𝑆)
4	3	cnveqi 5329	. . . . . . . 8 ⊢ ^◡^◡(𝑆 ∘ 𝑆) = ^◡(^◡𝑆 ∘ ^◡𝑆)
5		cnvco 5340	. . . . . . . 8 ⊢ ^◡(^◡𝑆 ∘ ^◡𝑆) = (^◡^◡𝑆 ∘ ^◡^◡𝑆)
6		cocnvcnv1 5684	. . . . . . . . 9 ⊢ (^◡^◡𝑆 ∘ ^◡^◡𝑆) = (𝑆 ∘ ^◡^◡𝑆)
7		cocnvcnv2 5685	. . . . . . . . 9 ⊢ (𝑆 ∘ ^◡^◡𝑆) = (𝑆 ∘ 𝑆)
8	6, 7	eqtri 2673	. . . . . . . 8 ⊢ (^◡^◡𝑆 ∘ ^◡^◡𝑆) = (𝑆 ∘ 𝑆)
9	4, 5, 8	3eqtri 2677	. . . . . . 7 ⊢ ^◡^◡(𝑆 ∘ 𝑆) = (𝑆 ∘ 𝑆)
10	9	sseq1i 3662	. . . . . 6 ⊢ (^◡^◡(𝑆 ∘ 𝑆) ⊆ ^◡^◡𝑆 ↔ (𝑆 ∘ 𝑆) ⊆ ^◡^◡𝑆)
11	10	biimpi 206	. . . . 5 ⊢ (^◡^◡(𝑆 ∘ 𝑆) ⊆ ^◡^◡𝑆 → (𝑆 ∘ 𝑆) ⊆ ^◡^◡𝑆)
12		cnvcnvss 5624	. . . . 5 ⊢ ^◡^◡𝑆 ⊆ 𝑆
13	11, 12	syl6ss 3648	. . . 4 ⊢ (^◡^◡(𝑆 ∘ 𝑆) ⊆ ^◡^◡𝑆 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
14	2, 13	syl 17	. . 3 ⊢ (^◡(𝑆 ∘ 𝑆) ⊆ ^◡𝑆 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
15	1, 14	impbii 199	. 2 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ ^◡(𝑆 ∘ 𝑆) ⊆ ^◡𝑆)
16	3	sseq1i 3662	. 2 ⊢ (^◡(𝑆 ∘ 𝑆) ⊆ ^◡𝑆 ↔ (^◡𝑆 ∘ ^◡𝑆) ⊆ ^◡𝑆)
17	15, 16	bitri 264	1 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ (^◡𝑆 ∘ ^◡𝑆) ⊆ ^◡𝑆)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 ⊆ wss 3607 ^◡ccnv 5142 ∘ ccom 5147

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155

This theorem is referenced by: (None)

W3C validator