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Theorem trclfvcotr 14903
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 6068 . . . . . . . . . 10 ((𝑟𝑟) ⊆ 𝑟 ↔ ∀𝑎𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
2 sp 2177 . . . . . . . . . . 11 (∀𝑎𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐) → ∀𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
3219.21bbi 2184 . . . . . . . . . 10 (∀𝑎𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐) → ((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
41, 3sylbi 216 . . . . . . . . 9 ((𝑟𝑟) ⊆ 𝑟 → ((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
54adantl 483 . . . . . . . 8 ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → ((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
65a2i 14 . . . . . . 7 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
76alimi 1814 . . . . . 6 (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
87ax-gen 1798 . . . . 5 𝑐(∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
98ax-gen 1798 . . . 4 𝑏𝑐(∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
109ax-gen 1798 . . 3 𝑎𝑏𝑐(∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
11 brtrclfv 14896 . . . . . . . 8 (𝑅𝑉 → (𝑎(t+‘𝑅)𝑏 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏)))
12 brtrclfv 14896 . . . . . . . 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 519 . . . . . . . . 9 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) ↔ (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
1514albii 1822 . . . . . . . 8 (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) ↔ ∀𝑟(((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
16 19.26 1874 . . . . . . . 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 14896 . . . . . 6 (𝑅𝑉 → (𝑎(t+‘𝑅)𝑐 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐)))
2018, 19imbi12d 345 . . . . 5 (𝑅𝑉 → (((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐) ↔ (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))))
2120albidv 1924 . . . 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 1927 . . 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 6068 . 2 (((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) ↔ ∀𝑎𝑏𝑐((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐))
2523, 24sylibr 233 1 (𝑅𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540  wcel 2107  wss 3914   class class class wbr 5109  ccom 5641  cfv 6500  t+ctcl 14879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-iota 6452  df-fun 6502  df-fv 6508  df-trcl 14881
This theorem is referenced by:  trclfvlb2  14904  trclidm  14907  trclfvcotrg  14910
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