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Theorem trclfvcotr 13976
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 5725 . . . . . . . . . 10 ((𝑟𝑟) ⊆ 𝑟 ↔ ∀𝑎𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
2 sp 2219 . . . . . . . . . . 11 (∀𝑎𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐) → ∀𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
3219.21bbi 2226 . . . . . . . . . 10 (∀𝑎𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐) → ((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
41, 3sylbi 208 . . . . . . . . 9 ((𝑟𝑟) ⊆ 𝑟 → ((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
54adantl 469 . . . . . . . 8 ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → ((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
65a2i 14 . . . . . . 7 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
76alimi 1896 . . . . . 6 (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
87ax-gen 1877 . . . . 5 𝑐(∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
98ax-gen 1877 . . . 4 𝑏𝑐(∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
109ax-gen 1877 . . 3 𝑎𝑏𝑐(∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
11 brtrclfv 13969 . . . . . . . 8 (𝑅𝑉 → (𝑎(t+‘𝑅)𝑏 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏)))
12 brtrclfv 13969 . . . . . . . 8 (𝑅𝑉 → (𝑏(t+‘𝑅)𝑐 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
1311, 12anbi12d 618 . . . . . . 7 (𝑅𝑉 → ((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) ↔ (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐))))
14 jcab 509 . . . . . . . . 9 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) ↔ (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
1514albii 1904 . . . . . . . 8 (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) ↔ ∀𝑟(((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
16 19.26 1959 . . . . . . . 8 (∀𝑟(((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)) ↔ (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
1715, 16bitri 266 . . . . . . 7 (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) ↔ (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
1813, 17syl6bbr 280 . . . . . 6 (𝑅𝑉 → ((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐))))
19 brtrclfv 13969 . . . . . 6 (𝑅𝑉 → (𝑎(t+‘𝑅)𝑐 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐)))
2018, 19imbi12d 335 . . . . 5 (𝑅𝑉 → (((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐) ↔ (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))))
2120albidv 2011 . . . 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 2014 . . 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 249 . 2 (𝑅𝑉 → ∀𝑎𝑏𝑐((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐))
24 cotr 5725 . 2 (((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) ↔ ∀𝑎𝑏𝑐((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐))
2523, 24sylibr 225 1 (𝑅𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1635  wcel 2157  wss 3776   class class class wbr 4851  ccom 5322  cfv 6104  t+ctcl 13952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-sbc 3641  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-int 4677  df-br 4852  df-opab 4914  df-mpt 4931  df-id 5226  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-iota 6067  df-fun 6106  df-fv 6112  df-trcl 13954
This theorem is referenced by:  trclfvlb2  13977  trclidm  13980  trclfvcotrg  13983
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