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Theorem trclfvcotrg 15052
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 15045 . 2 (𝑅 ∈ V → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
2 fvprc 6899 . . 3 𝑅 ∈ V → (t+‘𝑅) = ∅)
3 0trrel 15017 . . . . 5 (∅ ∘ ∅) ⊆ ∅
43a1i 11 . . . 4 ((t+‘𝑅) = ∅ → (∅ ∘ ∅) ⊆ ∅)
5 id 22 . . . . 5 ((t+‘𝑅) = ∅ → (t+‘𝑅) = ∅)
65, 5coeq12d 5878 . . . 4 ((t+‘𝑅) = ∅ → ((t+‘𝑅) ∘ (t+‘𝑅)) = (∅ ∘ ∅))
74, 6, 53sstr4d 4043 . . 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 1537  wcel 2106  Vcvv 3478  wss 3963  c0 4339  ccom 5693  cfv 6563  t+ctcl 15021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-trcl 15023
This theorem is referenced by:  cotrcltrcl  43715  brtrclfv2  43717  frege96d  43739  frege97d  43742  frege98d  43743  frege109d  43747  frege131d  43754
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