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Theorem trclfvcotr 14364
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 5943 . . . . . . . . . 10 ((𝑟𝑟) ⊆ 𝑟 ↔ ∀𝑎𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
2 sp 2181 . . . . . . . . . . 11 (∀𝑎𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐) → ∀𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
3219.21bbi 2188 . . . . . . . . . 10 (∀𝑎𝑏𝑐((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐) → ((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
41, 3sylbi 220 . . . . . . . . 9 ((𝑟𝑟) ⊆ 𝑟 → ((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
54adantl 485 . . . . . . . 8 ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → ((𝑎𝑟𝑏𝑏𝑟𝑐) → 𝑎𝑟𝑐))
65a2i 14 . . . . . . 7 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
76alimi 1813 . . . . . 6 (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
87ax-gen 1797 . . . . 5 𝑐(∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
98ax-gen 1797 . . . 4 𝑏𝑐(∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
109ax-gen 1797 . . 3 𝑎𝑏𝑐(∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))
11 brtrclfv 14357 . . . . . . . 8 (𝑅𝑉 → (𝑎(t+‘𝑅)𝑏 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏)))
12 brtrclfv 14357 . . . . . . . 8 (𝑅𝑉 → (𝑏(t+‘𝑅)𝑐 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
1311, 12anbi12d 633 . . . . . . 7 (𝑅𝑉 → ((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) ↔ (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐))))
14 jcab 521 . . . . . . . . 9 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) ↔ (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
1514albii 1821 . . . . . . . 8 (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) ↔ ∀𝑟(((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
16 19.26 1871 . . . . . . . 8 (∀𝑟(((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)) ↔ (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
1715, 16bitri 278 . . . . . . 7 (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) ↔ (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑏) ∧ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑏𝑟𝑐)))
1813, 17syl6bbr 292 . . . . . 6 (𝑅𝑉 → ((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐))))
19 brtrclfv 14357 . . . . . 6 (𝑅𝑉 → (𝑎(t+‘𝑅)𝑐 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐)))
2018, 19imbi12d 348 . . . . 5 (𝑅𝑉 → (((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐) ↔ (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑎𝑟𝑏𝑏𝑟𝑐)) → ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝑎𝑟𝑐))))
2120albidv 1921 . . . 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 1924 . . 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 261 . 2 (𝑅𝑉 → ∀𝑎𝑏𝑐((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐))
24 cotr 5943 . 2 (((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) ↔ ∀𝑎𝑏𝑐((𝑎(t+‘𝑅)𝑏𝑏(t+‘𝑅)𝑐) → 𝑎(t+‘𝑅)𝑐))
2523, 24sylibr 237 1 (𝑅𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536  wcel 2112  wss 3884   class class class wbr 5033  ccom 5527  cfv 6328  t+ctcl 14340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-iota 6287  df-fun 6330  df-fv 6336  df-trcl 14342
This theorem is referenced by:  trclfvlb2  14365  trclidm  14368  trclfvcotrg  14371
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