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Theorem trclidm 14195
 Description: The transitive closure of a relation is idempotent. (Contributed by RP, 29-Apr-2020.)
Assertion
Ref Expression
trclidm (𝑅𝑉 → (t+‘(t+‘𝑅)) = (t+‘𝑅))

Proof of Theorem trclidm
StepHypRef Expression
1 fvex 6543 . 2 (t+‘𝑅) ∈ V
2 trclfvcotr 14191 . 2 (𝑅𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
3 cotrtrclfv 14194 . 2 (((t+‘𝑅) ∈ V ∧ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) → (t+‘(t+‘𝑅)) = (t+‘𝑅))
41, 2, 3sylancr 587 1 (𝑅𝑉 → (t+‘(t+‘𝑅)) = (t+‘𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1520   ∈ wcel 2079  Vcvv 3432   ⊆ wss 3854   ∘ ccom 5439  ‘cfv 6217  t+ctcl 14167 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-int 4777  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-iota 6181  df-fun 6219  df-fv 6225  df-trcl 14169 This theorem is referenced by: (None)
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