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Theorem trclfvcotr 13598
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 5427 . . . . . . . . . 10 ((𝑟𝑟) ⊆ 𝑟 ↔ ∀𝑎𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
2 sp 2041 . . . . . . . . . . 11 (∀𝑎𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐) → ∀𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
3219.21bbi 2048 . . . . . . . . . 10 (∀𝑎𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐) → ((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
41, 3sylbi 206 . . . . . . . . 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 1730 . . . . . 6 (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
87ax-gen 1713 . . . . 5 𝑐(∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
98ax-gen 1713 . . . 4 𝑏𝑐(∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
109ax-gen 1713 . . 3 𝑎𝑏𝑐(∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
11 brtrclfv 13591 . . . . . . . 8 (𝑅𝑉 → (𝑎(t+‘𝑅)𝑏 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏)))
12 brtrclfv 13591 . . . . . . . 8 (𝑅𝑉 → (𝑏(t+‘𝑅)𝑐 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
1311, 12anbi12d 743 . . . . . . 7 (𝑅𝑉 → ((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) ↔ (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐))))
14 jcab 903 . . . . . . . . 9 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) ↔ (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
1514albii 1737 . . . . . . . 8 (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) ↔ ∀𝑟(((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
16 19.26 1786 . . . . . . . 8 (∀𝑟(((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)) ↔ (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
1715, 16bitri 263 . . . . . . 7 (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) ↔ (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
1813, 17syl6bbr 277 . . . . . 6 (𝑅𝑉 → ((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐))))
19 brtrclfv 13591 . . . . . 6 (𝑅𝑉 → (𝑎(t+‘𝑅)𝑐 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐)))
2018, 19imbi12d 333 . . . . 5 (𝑅𝑉 → (((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐) ↔ (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))))
2120albidv 1836 . . . 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 1838 . . 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 247 . 2 (𝑅𝑉 → ∀𝑎𝑏𝑐((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐))
24 cotr 5427 . 2 (((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) ↔ ∀𝑎𝑏𝑐((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐))
2523, 24sylibr 223 1 (𝑅𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1473  wcel 1977  wss 3540   class class class wbr 4583  ccom 5042  cfv 5804  t+ctcl 13572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-trcl 13574
This theorem is referenced by:  trclfvlb2  13599  trclidm  13602  trclfvcotrg  13605
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