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Theorem trclfvcotrg 14227
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 14220 . 2 (𝑅 ∈ V → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
2 fvprc 6486 . . 3 𝑅 ∈ V → (t+‘𝑅) = ∅)
3 0trrel 14192 . . . . 5 (∅ ∘ ∅) ⊆ ∅
43a1i 11 . . . 4 ((t+‘𝑅) = ∅ → (∅ ∘ ∅) ⊆ ∅)
5 id 22 . . . . 5 ((t+‘𝑅) = ∅ → (t+‘𝑅) = ∅)
65, 5coeq12d 5578 . . . 4 ((t+‘𝑅) = ∅ → ((t+‘𝑅) ∘ (t+‘𝑅)) = (∅ ∘ ∅))
74, 6, 53sstr4d 3900 . . 3 ((t+‘𝑅) = ∅ → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
82, 7syl 17 . 2 𝑅 ∈ V → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
91, 8pm2.61i 177 1 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1507  wcel 2048  Vcvv 3409  wss 3825  c0 4173  ccom 5404  cfv 6182  t+ctcl 14196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-int 4744  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-iota 6146  df-fun 6184  df-fv 6190  df-trcl 14198
This theorem is referenced by:  cotrcltrcl  39378  brtrclfv2  39380  frege96d  39402  frege97d  39405  frege98d  39406  frege109d  39410  frege131d  39417
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