Theorem relcnvtr 5655

Description: A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)

Assertion

Ref	Expression
relcnvtr	⊢ (Rel 𝑅 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅))

Proof of Theorem relcnvtr

Step	Hyp	Ref	Expression
1		cnvco 5308	. . 3 ⊢ ◡(𝑅 ∘ 𝑅) = (◡𝑅 ∘ ◡𝑅)
2		cnvss 5294	. . 3 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅)
3	1, 2	syl5eqssr 3650	. 2 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅)
4		cnvco 5308	. . . 4 ⊢ ◡(◡𝑅 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑅)
5		cnvss 5294	. . . 4 ⊢ ((◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅 → ◡(◡𝑅 ∘ ◡𝑅) ⊆ ◡◡𝑅)
6		sseq1 3626	. . . . 5 ⊢ (◡(◡𝑅 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑅) → (◡(◡𝑅 ∘ ◡𝑅) ⊆ ◡◡𝑅 ↔ (◡◡𝑅 ∘ ◡◡𝑅) ⊆ ◡◡𝑅))
7		dfrel2 5583	. . . . . . 7 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
8		coeq1 5279	. . . . . . . . . 10 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∘ ◡◡𝑅) = (𝑅 ∘ ◡◡𝑅))
9		coeq2 5280	. . . . . . . . . 10 ⊢ (◡◡𝑅 = 𝑅 → (𝑅 ∘ ◡◡𝑅) = (𝑅 ∘ 𝑅))
10	8, 9	eqtrd 2656	. . . . . . . . 9 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∘ ◡◡𝑅) = (𝑅 ∘ 𝑅))
11		id 22	. . . . . . . . 9 ⊢ (◡◡𝑅 = 𝑅 → ◡◡𝑅 = 𝑅)
12	10, 11	sseq12d 3634	. . . . . . . 8 ⊢ (◡◡𝑅 = 𝑅 → ((◡◡𝑅 ∘ ◡◡𝑅) ⊆ ◡◡𝑅 ↔ (𝑅 ∘ 𝑅) ⊆ 𝑅))
13	12	biimpd 219	. . . . . . 7 ⊢ (◡◡𝑅 = 𝑅 → ((◡◡𝑅 ∘ ◡◡𝑅) ⊆ ◡◡𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅))
14	7, 13	sylbi 207	. . . . . 6 ⊢ (Rel 𝑅 → ((◡◡𝑅 ∘ ◡◡𝑅) ⊆ ◡◡𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅))
15	14	com12 32	. . . . 5 ⊢ ((◡◡𝑅 ∘ ◡◡𝑅) ⊆ ◡◡𝑅 → (Rel 𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅))
16	6, 15	syl6bi 243	. . . 4 ⊢ (◡(◡𝑅 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑅) → (◡(◡𝑅 ∘ ◡𝑅) ⊆ ◡◡𝑅 → (Rel 𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅)))
17	4, 5, 16	mpsyl 68	. . 3 ⊢ ((◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅 → (Rel 𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅))
18	17	com12 32	. 2 ⊢ (Rel 𝑅 → ((◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅))
19	3, 18	impbid2 216	1 ⊢ (Rel 𝑅 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ⊆ wss 3574  ^◡ccnv 5113   ∘ ccom 5118  Rel wrel 5119

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-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

This theorem is referenced by: (None)