Theorem trclfvcotr 15048

Description: The transitive closure of a relation is a transitive relation. (Contributed by RP, 29-Apr-2020.)

Assertion

Ref	Expression
trclfvcotr	⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))

Proof of Theorem trclfvcotr

Dummy variables 𝑎 𝑏 𝑐 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cotr 6130	. . . . . . . . . 10 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ ∀𝑎∀𝑏∀𝑐((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐))
2		sp 2183	. . . . . . . . . . 11 ⊢ (∀𝑎∀𝑏∀𝑐((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐) → ∀𝑏∀𝑐((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐))
3	2	19.21bbi 2190	. . . . . . . . . 10 ⊢ (∀𝑎∀𝑏∀𝑐((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐) → ((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐))
4	1, 3	sylbi 217	. . . . . . . . 9 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 → ((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐))
5	4	adantl 481	. . . . . . . 8 ⊢ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → ((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐))
6	5	a2i 14	. . . . . . 7 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐)) → ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
7	6	alimi 1811	. . . . . 6 ⊢ (∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐)) → ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
8	7	ax-gen 1795	. . . . 5 ⊢ ∀𝑐(∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐)) → ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
9	8	ax-gen 1795	. . . 4 ⊢ ∀𝑏∀𝑐(∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐)) → ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
10	9	ax-gen 1795	. . 3 ⊢ ∀𝑎∀𝑏∀𝑐(∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐)) → ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
11		brtrclfv 15041	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑎(t+‘𝑅)𝑏 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏)))
12		brtrclfv 15041	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑏(t+‘𝑅)𝑐 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
13	11, 12	anbi12d 632	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ((𝑎(t+‘𝑅)𝑏 ∧ 𝑏(t+‘𝑅)𝑐) ↔ (∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐))))
14		jcab 517	. . . . . . . . 9 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐)) ↔ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
15	14	albii 1819	. . . . . . . 8 ⊢ (∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐)) ↔ ∀𝑟(((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
16		19.26 1870	. . . . . . . 8 ⊢ (∀𝑟(((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)) ↔ (∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
17	15, 16	bitri 275	. . . . . . 7 ⊢ (∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐)) ↔ (∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
18	13, 17	bitr4di 289	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ((𝑎(t+‘𝑅)𝑏 ∧ 𝑏(t+‘𝑅)𝑐) ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐))))
19		brtrclfv 15041	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑎(t+‘𝑅)𝑐 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐)))
20	18, 19	imbi12d 344	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (((𝑎(t+‘𝑅)𝑏 ∧ 𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐) ↔ (∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐)) → ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))))
21	20	albidv 1920	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (∀𝑐((𝑎(t+‘𝑅)𝑏 ∧ 𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐) ↔ ∀𝑐(∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐)) → ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))))
22	21	2albidv 1923	. . 3 ⊢ (𝑅 ∈ 𝑉 → (∀𝑎∀𝑏∀𝑐((𝑎(t+‘𝑅)𝑏 ∧ 𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐) ↔ ∀𝑎∀𝑏∀𝑐(∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐)) → ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))))
23	10, 22	mpbiri 258	. 2 ⊢ (𝑅 ∈ 𝑉 → ∀𝑎∀𝑏∀𝑐((𝑎(t+‘𝑅)𝑏 ∧ 𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐))
24		cotr 6130	. 2 ⊢ (((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) ↔ ∀𝑎∀𝑏∀𝑐((𝑎(t+‘𝑅)𝑏 ∧ 𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐))
25	23, 24	sylibr 234	1 ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   ∈ wcel 2108   ⊆ wss 3951   class class class wbr 5143   ∘ ccom 5689  ‘cfv 6561  t+ctcl 15024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-trcl 15026

This theorem is referenced by:  trclfvlb2  15049  trclidm  15052  trclfvcotrg  15055