cnvtrrel - Mathbox for Richard Penner

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Theorem cnvtrrel 38745

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 5498	. . 3 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 → ^◡(𝑆 ∘ 𝑆) ⊆ ^◡𝑆)
2		cnvss 5498	. . . 4 ⊢ (^◡(𝑆 ∘ 𝑆) ⊆ ^◡𝑆 → ^◡^◡(𝑆 ∘ 𝑆) ⊆ ^◡^◡𝑆)
3		cnvco 5511	. . . . . . . . 9 ⊢ ^◡(𝑆 ∘ 𝑆) = (^◡𝑆 ∘ ^◡𝑆)
4	3	cnveqi 5500	. . . . . . . 8 ⊢ ^◡^◡(𝑆 ∘ 𝑆) = ^◡(^◡𝑆 ∘ ^◡𝑆)
5		cnvco 5511	. . . . . . . 8 ⊢ ^◡(^◡𝑆 ∘ ^◡𝑆) = (^◡^◡𝑆 ∘ ^◡^◡𝑆)
6		cocnvcnv1 5865	. . . . . . . . 9 ⊢ (^◡^◡𝑆 ∘ ^◡^◡𝑆) = (𝑆 ∘ ^◡^◡𝑆)
7		cocnvcnv2 5866	. . . . . . . . 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 3825	. . . . . 6 ⊢ (^◡^◡(𝑆 ∘ 𝑆) ⊆ ^◡^◡𝑆 ↔ (𝑆 ∘ 𝑆) ⊆ ^◡^◡𝑆)
11	10	biimpi 208	. . . . 5 ⊢ (^◡^◡(𝑆 ∘ 𝑆) ⊆ ^◡^◡𝑆 → (𝑆 ∘ 𝑆) ⊆ ^◡^◡𝑆)
12		cnvcnvss 5805	. . . . 5 ⊢ ^◡^◡𝑆 ⊆ 𝑆
13	11, 12	syl6ss 3810	. . . 4 ⊢ (^◡^◡(𝑆 ∘ 𝑆) ⊆ ^◡^◡𝑆 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
14	2, 13	syl 17	. . 3 ⊢ (^◡(𝑆 ∘ 𝑆) ⊆ ^◡𝑆 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
15	1, 14	impbii 201	. 2 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ ^◡(𝑆 ∘ 𝑆) ⊆ ^◡𝑆)
16	3	sseq1i 3825	. 2 ⊢ (^◡(𝑆 ∘ 𝑆) ⊆ ^◡𝑆 ↔ (^◡𝑆 ∘ ^◡𝑆) ⊆ ^◡𝑆)
17	15, 16	bitri 267	1 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ (^◡𝑆 ∘ ^◡𝑆) ⊆ ^◡𝑆)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 198 ⊆ wss 3769 ^◡ccnv 5311 ∘ ccom 5316

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324

This theorem is referenced by: (None)

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