Theorem cnvtrrel 43989

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 5822	. . 3 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 → ◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆)
2		cnvss 5822	. . . 4 ⊢ (◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆 → ◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆)
3		cnvco 5835	. . . . . . . . 9 ⊢ ◡(𝑆 ∘ 𝑆) = (◡𝑆 ∘ ◡𝑆)
4	3	cnveqi 5824	. . . . . . . 8 ⊢ ◡◡(𝑆 ∘ 𝑆) = ◡(◡𝑆 ∘ ◡𝑆)
5		cnvco 5835	. . . . . . . 8 ⊢ ◡(◡𝑆 ∘ ◡𝑆) = (◡◡𝑆 ∘ ◡◡𝑆)
6		cocnvcnv1 6217	. . . . . . . . 9 ⊢ (◡◡𝑆 ∘ ◡◡𝑆) = (𝑆 ∘ ◡◡𝑆)
7		cocnvcnv2 6218	. . . . . . . . 9 ⊢ (𝑆 ∘ ◡◡𝑆) = (𝑆 ∘ 𝑆)
8	6, 7	eqtri 2760	. . . . . . . 8 ⊢ (◡◡𝑆 ∘ ◡◡𝑆) = (𝑆 ∘ 𝑆)
9	4, 5, 8	3eqtri 2764	. . . . . . 7 ⊢ ◡◡(𝑆 ∘ 𝑆) = (𝑆 ∘ 𝑆)
10	9	sseq1i 3963	. . . . . 6 ⊢ (◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆 ↔ (𝑆 ∘ 𝑆) ⊆ ◡◡𝑆)
11	10	biimpi 216	. . . . 5 ⊢ (◡◡(𝑆 ∘ 𝑆) ⊆ ◡◡𝑆 → (𝑆 ∘ 𝑆) ⊆ ◡◡𝑆)
12		cnvcnvss 6153	. . . . 5 ⊢ ◡◡𝑆 ⊆ 𝑆
13	11, 12	sstrdi 3947	. . . 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 3963	. 2 ⊢ (◡(𝑆 ∘ 𝑆) ⊆ ◡𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆)
17	15, 16	bitri 275	1 ⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆)

Syntax hints:   ↔ wb 206   ⊆ wss 3902  ^◡ccnv 5624   ∘ ccom 5629

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637

This theorem is referenced by: (None)