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Theorem trclfvcotrg 14958
Description: The value of the transitive closure of a relation is always a transitive relation. (Contributed by RP, 8-May-2020.)
Assertion
Ref Expression
trclfvcotrg ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)

Proof of Theorem trclfvcotrg
StepHypRef Expression
1 trclfvcotr 14951 . 2 (𝑅 ∈ V → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
2 fvprc 6879 . . 3 𝑅 ∈ V → (t+‘𝑅) = ∅)
3 0trrel 14923 . . . . 5 (∅ ∘ ∅) ⊆ ∅
43a1i 11 . . . 4 ((t+‘𝑅) = ∅ → (∅ ∘ ∅) ⊆ ∅)
5 id 22 . . . . 5 ((t+‘𝑅) = ∅ → (t+‘𝑅) = ∅)
65, 5coeq12d 5861 . . . 4 ((t+‘𝑅) = ∅ → ((t+‘𝑅) ∘ (t+‘𝑅)) = (∅ ∘ ∅))
74, 6, 53sstr4d 4027 . . 3 ((t+‘𝑅) = ∅ → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
82, 7syl 17 . 2 𝑅 ∈ V → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
91, 8pm2.61i 182 1 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3475  wss 3946  c0 4320  ccom 5678  cfv 6539  t+ctcl 14927
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 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-int 4949  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-iota 6491  df-fun 6541  df-fv 6547  df-trcl 14929
This theorem is referenced by:  cotrcltrcl  42408  brtrclfv2  42410  frege96d  42432  frege97d  42435  frege98d  42436  frege109d  42440  frege131d  42447
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