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Theorem trclfvcotrg 14981
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 14974 . 2 (𝑅 ∈ V → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
2 fvprc 6883 . . 3 𝑅 ∈ V → (t+‘𝑅) = ∅)
3 0trrel 14946 . . . . 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 4025 . . 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 1534  wcel 2099  Vcvv 3469  wss 3944  c0 4318  ccom 5676  cfv 6542  t+ctcl 14950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6494  df-fun 6544  df-fv 6550  df-trcl 14952
This theorem is referenced by:  cotrcltrcl  43068  brtrclfv2  43070  frege96d  43092  frege97d  43095  frege98d  43096  frege109d  43100  frege131d  43107
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