Theorem relcnvtr 5190

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 4852	. . 3 ⊢ ◡(𝑅 ∘ 𝑅) = (◡𝑅 ∘ ◡𝑅)
2		cnvss 4840	. . 3 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅)
3	1, 2	eqsstrrid 3231	. 2 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅)
4		cnvco 4852	. . . 4 ⊢ ◡(◡𝑅 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑅)
5		cnvss 4840	. . . 4 ⊢ ((◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅 → ◡(◡𝑅 ∘ ◡𝑅) ⊆ ◡◡𝑅)
6		sseq1 3207	. . . . 5 ⊢ (◡(◡𝑅 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑅) → (◡(◡𝑅 ∘ ◡𝑅) ⊆ ◡◡𝑅 ↔ (◡◡𝑅 ∘ ◡◡𝑅) ⊆ ◡◡𝑅))
7		dfrel2 5121	. . . . . . 7 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
8		coeq1 4824	. . . . . . . . . 10 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∘ ◡◡𝑅) = (𝑅 ∘ ◡◡𝑅))
9		coeq2 4825	. . . . . . . . . 10 ⊢ (◡◡𝑅 = 𝑅 → (𝑅 ∘ ◡◡𝑅) = (𝑅 ∘ 𝑅))
10	8, 9	eqtrd 2229	. . . . . . . . 9 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∘ ◡◡𝑅) = (𝑅 ∘ 𝑅))
11		id 19	. . . . . . . . 9 ⊢ (◡◡𝑅 = 𝑅 → ◡◡𝑅 = 𝑅)
12	10, 11	sseq12d 3215	. . . . . . . 8 ⊢ (◡◡𝑅 = 𝑅 → ((◡◡𝑅 ∘ ◡◡𝑅) ⊆ ◡◡𝑅 ↔ (𝑅 ∘ 𝑅) ⊆ 𝑅))
13	12	biimpd 144	. . . . . . 7 ⊢ (◡◡𝑅 = 𝑅 → ((◡◡𝑅 ∘ ◡◡𝑅) ⊆ ◡◡𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅))
14	7, 13	sylbi 121	. . . . . 6 ⊢ (Rel 𝑅 → ((◡◡𝑅 ∘ ◡◡𝑅) ⊆ ◡◡𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅))
15	14	com12 30	. . . . 5 ⊢ ((◡◡𝑅 ∘ ◡◡𝑅) ⊆ ◡◡𝑅 → (Rel 𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅))
16	6, 15	biimtrdi 163	. . . 4 ⊢ (◡(◡𝑅 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑅) → (◡(◡𝑅 ∘ ◡𝑅) ⊆ ◡◡𝑅 → (Rel 𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅)))
17	4, 5, 16	mpsyl 65	. . 3 ⊢ ((◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅 → (Rel 𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅))
18	17	com12 30	. 2 ⊢ (Rel 𝑅 → ((◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅))
19	3, 18	impbid2 143	1 ⊢ (Rel 𝑅 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364   ⊆ wss 3157  ^◡ccnv 4663   ∘ ccom 4668  Rel wrel 4669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673

This theorem is referenced by: (None)