Theorem trclfvcotr 15205

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 6124	. . . . . . . . . 10 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ ∀𝑎∀𝑏∀𝑐((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐))
2		sp 2223	. . . . . . . . . . 11 ⊢ (∀𝑎∀𝑏∀𝑐((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐) → ∀𝑏∀𝑐((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐))
3	2	19.21bbi 2230	. . . . . . . . . 10 ⊢ (∀𝑎∀𝑏∀𝑐((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐) → ((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐))
4	1, 3	sylbi 220	. . . . . . . . 9 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 → ((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐))
5	4	adantl 487	. . . . . . . 8 ⊢ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → ((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐))
6	5	a2i 15	. . . . . . 7 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐)) → ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
7	6	alimi 1844	. . . . . 6 ⊢ (∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐)) → ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
8	7	ax-gen 1828	. . . . 5 ⊢ ∀𝑐(∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐)) → ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
9	8	ax-gen 1828	. . . 4 ⊢ ∀𝑏∀𝑐(∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐)) → ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
10	9	ax-gen 1828	. . 3 ⊢ ∀𝑎∀𝑏∀𝑐(∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐)) → ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
11		brtrclfv 15198	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑎(t+‘𝑅)𝑏 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏)))
12		brtrclfv 15198	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑏(t+‘𝑅)𝑐 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
13	11, 12	anbi12d 644	. . . . . . 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 527	. . . . . . . . 9 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐)) ↔ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
15	14	albii 1852	. . . . . . . 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 1903	. . . . . . . 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 278	. . . . . . 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 292	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ((𝑎(t+‘𝑅)𝑏 ∧ 𝑏(t+‘𝑅)𝑐) ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐))))
19		brtrclfv 15198	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑎(t+‘𝑅)𝑐 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐)))
20	18, 19	imbi12d 347	. . . . 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 1955	. . . 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 1958	. . 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 261	. 2 ⊢ (𝑅 ∈ 𝑉 → ∀𝑎∀𝑏∀𝑐((𝑎(t+‘𝑅)𝑏 ∧ 𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐))
24		cotr 6124	. 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 237	1 ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   ∈ wcel 2147   ⊆ wss 3900   class class class wbr 5108   ∘ ccom 5662  ‘cfv 6566  t+ctcl 15181

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-iota 6520  df-fun 6568  df-fv 6574  df-trcl 15183

This theorem is used by:  trclfvlb2  15206  trclidm  15209  trclfvcotrg  15212