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Mirrors > Home > MPE Home > Th. List > trclidm | Structured version Visualization version GIF version |
Description: The transitive closure of a relation is idempotent. (Contributed by RP, 29-Apr-2020.) |
Ref | Expression |
---|---|
trclidm | ⊢ (𝑅 ∈ 𝑉 → (t+‘(t+‘𝑅)) = (t+‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6787 | . 2 ⊢ (t+‘𝑅) ∈ V | |
2 | trclfvcotr 14720 | . 2 ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) | |
3 | cotrtrclfv 14723 | . 2 ⊢ (((t+‘𝑅) ∈ V ∧ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) → (t+‘(t+‘𝑅)) = (t+‘𝑅)) | |
4 | 1, 2, 3 | sylancr 587 | 1 ⊢ (𝑅 ∈ 𝑉 → (t+‘(t+‘𝑅)) = (t+‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ⊆ wss 3887 ∘ ccom 5593 ‘cfv 6433 t+ctcl 14696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-iota 6391 df-fun 6435 df-fv 6441 df-trcl 14698 |
This theorem is referenced by: (None) |
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