Theorem relcnvtr 4840

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 4520	. . 3 ⊢ ◡(𝑅 ∘ 𝑅) = (◡𝑅 ∘ ◡𝑅)
2		cnvss 4508	. . 3 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅)
3	1, 2	syl5eqssr 2990	. 2 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅)
4		cnvco 4520	. . . 4 ⊢ ◡(◡𝑅 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑅)
5		cnvss 4508	. . . 4 ⊢ ((◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅 → ◡(◡𝑅 ∘ ◡𝑅) ⊆ ◡◡𝑅)
6		sseq1 2966	. . . . 5 ⊢ (◡(◡𝑅 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑅) → (◡(◡𝑅 ∘ ◡𝑅) ⊆ ◡◡𝑅 ↔ (◡◡𝑅 ∘ ◡◡𝑅) ⊆ ◡◡𝑅))
7		dfrel2 4771	. . . . . . 7 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
8		coeq1 4493	. . . . . . . . . 10 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∘ ◡◡𝑅) = (𝑅 ∘ ◡◡𝑅))
9		coeq2 4494	. . . . . . . . . 10 ⊢ (◡◡𝑅 = 𝑅 → (𝑅 ∘ ◡◡𝑅) = (𝑅 ∘ 𝑅))
10	8, 9	eqtrd 2072	. . . . . . . . 9 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∘ ◡◡𝑅) = (𝑅 ∘ 𝑅))
11		id 19	. . . . . . . . 9 ⊢ (◡◡𝑅 = 𝑅 → ◡◡𝑅 = 𝑅)
12	10, 11	sseq12d 2974	. . . . . . . 8 ⊢ (◡◡𝑅 = 𝑅 → ((◡◡𝑅 ∘ ◡◡𝑅) ⊆ ◡◡𝑅 ↔ (𝑅 ∘ 𝑅) ⊆ 𝑅))
13	12	biimpd 132	. . . . . . 7 ⊢ (◡◡𝑅 = 𝑅 → ((◡◡𝑅 ∘ ◡◡𝑅) ⊆ ◡◡𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅))
14	7, 13	sylbi 114	. . . . . 6 ⊢ (Rel 𝑅 → ((◡◡𝑅 ∘ ◡◡𝑅) ⊆ ◡◡𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅))
15	14	com12 27	. . . . 5 ⊢ ((◡◡𝑅 ∘ ◡◡𝑅) ⊆ ◡◡𝑅 → (Rel 𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅))
16	6, 15	syl6bi 152	. . . 4 ⊢ (◡(◡𝑅 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑅) → (◡(◡𝑅 ∘ ◡𝑅) ⊆ ◡◡𝑅 → (Rel 𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅)))
17	4, 5, 16	mpsyl 59	. . 3 ⊢ ((◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅 → (Rel 𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅))
18	17	com12 27	. 2 ⊢ (Rel 𝑅 → ((◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅))
19	3, 18	impbid2 131	1 ⊢ (Rel 𝑅 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ⊆ wss 2917  ^◡ccnv 4344   ∘ ccom 4349  Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354

This theorem is referenced by: (None)