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Theorem trclfvcotrg 14369
 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 14362 . 2 (𝑅 ∈ V → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
2 fvprc 6638 . . 3 𝑅 ∈ V → (t+‘𝑅) = ∅)
3 0trrel 14334 . . . . 5 (∅ ∘ ∅) ⊆ ∅
43a1i 11 . . . 4 ((t+‘𝑅) = ∅ → (∅ ∘ ∅) ⊆ ∅)
5 id 22 . . . . 5 ((t+‘𝑅) = ∅ → (t+‘𝑅) = ∅)
65, 5coeq12d 5699 . . . 4 ((t+‘𝑅) = ∅ → ((t+‘𝑅) ∘ (t+‘𝑅)) = (∅ ∘ ∅))
74, 6, 53sstr4d 3962 . . 3 ((t+‘𝑅) = ∅ → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
82, 7syl 17 . 2 𝑅 ∈ V → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
91, 8pm2.61i 185 1 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881  ∅c0 4243   ∘ ccom 5523  ‘cfv 6324  t+ctcl 14338 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fv 6332  df-trcl 14340 This theorem is referenced by:  cotrcltrcl  40441  brtrclfv2  40443  frege96d  40465  frege97d  40468  frege98d  40469  frege109d  40473  frege131d  40480
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