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Theorem trclfvcotr 15048
Description: The transitive closure of a relation is a transitive relation. (Contributed by RP, 29-Apr-2020.)
Assertion
Ref Expression
trclfvcotr (𝑅𝑉 → ((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.
StepHypRef Expression
1 cotr 6130 . . . . . . . . . 10 ((𝑟𝑟) ⊆ 𝑟 ↔ ∀𝑎𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
2 sp 2183 . . . . . . . . . . 11 (∀𝑎𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐) → ∀𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
3219.21bbi 2190 . . . . . . . . . 10 (∀𝑎𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐) → ((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
41, 3sylbi 217 . . . . . . . . 9 ((𝑟𝑟) ⊆ 𝑟 → ((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
54adantl 481 . . . . . . . 8 ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → ((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
65a2i 14 . . . . . . 7 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
76alimi 1811 . . . . . 6 (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
87ax-gen 1795 . . . . 5 𝑐(∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
98ax-gen 1795 . . . 4 𝑏𝑐(∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
109ax-gen 1795 . . 3 𝑎𝑏𝑐(∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
11 brtrclfv 15041 . . . . . . . 8 (𝑅𝑉 → (𝑎(t+‘𝑅)𝑏 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏)))
12 brtrclfv 15041 . . . . . . . 8 (𝑅𝑉 → (𝑏(t+‘𝑅)𝑐 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
1311, 12anbi12d 632 . . . . . . 7 (𝑅𝑉 → ((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) ↔ (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐))))
14 jcab 517 . . . . . . . . 9 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) ↔ (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
1514albii 1819 . . . . . . . 8 (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) ↔ ∀𝑟(((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
16 19.26 1870 . . . . . . . 8 (∀𝑟(((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)) ↔ (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
1715, 16bitri 275 . . . . . . 7 (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) ↔ (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
1813, 17bitr4di 289 . . . . . 6 (𝑅𝑉 → ((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐))))
19 brtrclfv 15041 . . . . . 6 (𝑅𝑉 → (𝑎(t+‘𝑅)𝑐 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐)))
2018, 19imbi12d 344 . . . . 5 (𝑅𝑉 → (((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐) ↔ (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))))
2120albidv 1920 . . . 4 (𝑅𝑉 → (∀𝑐((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐) ↔ ∀𝑐(∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))))
22212albidv 1923 . . 3 (𝑅𝑉 → (∀𝑎𝑏𝑐((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐) ↔ ∀𝑎𝑏𝑐(∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))))
2310, 22mpbiri 258 . 2 (𝑅𝑉 → ∀𝑎𝑏𝑐((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐))
24 cotr 6130 . 2 (((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) ↔ ∀𝑎𝑏𝑐((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐))
2523, 24sylibr 234 1 (𝑅𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
Colors of variables: wff setvar class
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
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