Theorem relcnvtr 5284

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 4942	. . 3 ⊢ ◡(𝑅 ∘ 𝑅) = (◡𝑅 ∘ ◡𝑅)
2		cnvss 4930	. . 3 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅)
3	1, 2	eqsstrrid 3287	. 2 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅)
4		cnvco 4942	. . . 4 ⊢ ◡(◡𝑅 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑅)
5		cnvss 4930	. . . 4 ⊢ ((◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅 → ◡(◡𝑅 ∘ ◡𝑅) ⊆ ◡◡𝑅)
6		sseq1 3263	. . . . 5 ⊢ (◡(◡𝑅 ∘ ◡𝑅) = (◡◡𝑅 ∘ ◡◡𝑅) → (◡(◡𝑅 ∘ ◡𝑅) ⊆ ◡◡𝑅 ↔ (◡◡𝑅 ∘ ◡◡𝑅) ⊆ ◡◡𝑅))
7		dfrel2 5215	. . . . . . 7 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
8		coeq1 4914	. . . . . . . . . 10 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∘ ◡◡𝑅) = (𝑅 ∘ ◡◡𝑅))
9		coeq2 4915	. . . . . . . . . 10 ⊢ (◡◡𝑅 = 𝑅 → (𝑅 ∘ ◡◡𝑅) = (𝑅 ∘ 𝑅))
10	8, 9	eqtrd 2267	. . . . . . . . 9 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∘ ◡◡𝑅) = (𝑅 ∘ 𝑅))
11		id 19	. . . . . . . . 9 ⊢ (◡◡𝑅 = 𝑅 → ◡◡𝑅 = 𝑅)
12	10, 11	sseq12d 3271	. . . . . . . 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 1398   ⊆ wss 3213  ^◡ccnv 4750   ∘ ccom 4755  Rel wrel 4756

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760

This theorem is referenced by: (None)