Theorem trclfvcotr 15065

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 6114	. . . . . . . . . 10 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ ∀𝑎∀𝑏∀𝑐((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐))
2		sp 2222	. . . . . . . . . . 11 ⊢ (∀𝑎∀𝑏∀𝑐((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐) → ∀𝑏∀𝑐((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐))
3	2	19.21bbi 2229	. . . . . . . . . 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 15058	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑎(t+‘𝑅)𝑏 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏)))
12		brtrclfv 15058	. . . . . . . 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 15058	. . . . . 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 1953	. . . 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 1956	. . 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 6114	. 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 2146   ⊆ wss 3906   class class class wbr 5111   ∘ ccom 5667  ‘cfv 6540  t+ctcl 15041

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pow 5338  ax-pr 5406  ax-un 7738

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-iota 6496  df-fun 6542  df-fv 6548  df-trcl 15043

This theorem is used by:  trclfvlb2  15066  trclidm  15069  trclfvcotrg  15072