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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 6543 | . 2 ⊢ (t+‘𝑅) ∈ V | |
2 | trclfvcotr 14191 | . 2 ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) | |
3 | cotrtrclfv 14194 | . 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 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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