cnvtrrel - Mathbox for Richard Penner

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Theorem cnvtrrel 37962

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 5294	. . 3 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 → ^◡(𝑆 ∘ 𝑆) ⊆ ^◡𝑆)
2		cnvss 5294	. . . 4 ⊢ (^◡(𝑆 ∘ 𝑆) ⊆ ^◡𝑆 → ^◡^◡(𝑆 ∘ 𝑆) ⊆ ^◡^◡𝑆)
3		cnvco 5308	. . . . . . . . 9 ⊢ ^◡(𝑆 ∘ 𝑆) = (^◡𝑆 ∘ ^◡𝑆)
4	3	cnveqi 5297	. . . . . . . 8 ⊢ ^◡^◡(𝑆 ∘ 𝑆) = ^◡(^◡𝑆 ∘ ^◡𝑆)
5		cnvco 5308	. . . . . . . 8 ⊢ ^◡(^◡𝑆 ∘ ^◡𝑆) = (^◡^◡𝑆 ∘ ^◡^◡𝑆)
6		cocnvcnv1 5646	. . . . . . . . 9 ⊢ (^◡^◡𝑆 ∘ ^◡^◡𝑆) = (𝑆 ∘ ^◡^◡𝑆)
7		cocnvcnv2 5647	. . . . . . . . 9 ⊢ (𝑆 ∘ ^◡^◡𝑆) = (𝑆 ∘ 𝑆)
8	6, 7	eqtri 2644	. . . . . . . 8 ⊢ (^◡^◡𝑆 ∘ ^◡^◡𝑆) = (𝑆 ∘ 𝑆)
9	4, 5, 8	3eqtri 2648	. . . . . . 7 ⊢ ^◡^◡(𝑆 ∘ 𝑆) = (𝑆 ∘ 𝑆)
10	9	sseq1i 3629	. . . . . 6 ⊢ (^◡^◡(𝑆 ∘ 𝑆) ⊆ ^◡^◡𝑆 ↔ (𝑆 ∘ 𝑆) ⊆ ^◡^◡𝑆)
11	10	biimpi 206	. . . . 5 ⊢ (^◡^◡(𝑆 ∘ 𝑆) ⊆ ^◡^◡𝑆 → (𝑆 ∘ 𝑆) ⊆ ^◡^◡𝑆)
12		cnvcnvss 5589	. . . . 5 ⊢ ^◡^◡𝑆 ⊆ 𝑆
13	11, 12	syl6ss 3615	. . . 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 3629	. 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 3574 ^◡ccnv 5113 ∘ ccom 5118

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126

This theorem is referenced by: (None)

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