Theorem trclfvcotr 14960

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 6110	. . . . . . . . . 10 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ ∀𝑎∀𝑏∀𝑐((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐))
2		sp 2174	. . . . . . . . . . 11 ⊢ (∀𝑎∀𝑏∀𝑐((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐) → ∀𝑏∀𝑐((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐))
3	2	19.21bbi 2181	. . . . . . . . . 10 ⊢ (∀𝑎∀𝑏∀𝑐((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐) → ((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐))
4	1, 3	sylbi 216	. . . . . . . . 9 ⊢ ((𝑟 ∘ 𝑟) ⊆ 𝑟 → ((𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐) → 𝑎𝑟𝑐))
5	4	adantl 480	. . . . . . . 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 14953	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑎(t+‘𝑅)𝑏 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏)))
12		brtrclfv 14953	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑏(t+‘𝑅)𝑐 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
13	11, 12	anbi12d 629	. . . . . . 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 516	. . . . . . . . 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 1871	. . . . . . . 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 274	. . . . . . 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 288	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ((𝑎(t+‘𝑅)𝑏 ∧ 𝑏(t+‘𝑅)𝑐) ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏 ∧ 𝑏𝑟𝑐))))
19		brtrclfv 14953	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑎(t+‘𝑅)𝑐 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐)))
20	18, 19	imbi12d 343	. . . . 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 1921	. . . 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 1924	. . 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 257	. 2 ⊢ (𝑅 ∈ 𝑉 → ∀𝑎∀𝑏∀𝑐((𝑎(t+‘𝑅)𝑏 ∧ 𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐))
24		cotr 6110	. 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 233	1 ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))

Syntax hints:   → wi 4   ∧ wa 394  ∀wal 1537   ∈ wcel 2104   ⊆ wss 3947   class class class wbr 5147   ∘ ccom 5679  ‘cfv 6542  t+ctcl 14936

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6494  df-fun 6544  df-fv 6550  df-trcl 14938

This theorem is referenced by:  trclfvlb2  14961  trclidm  14964  trclfvcotrg  14967