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Mirrors > Home > MPE Home > Th. List > trclfvcotrg | Structured version Visualization version GIF version |
Description: The value of the transitive closure of a relation is always a transitive relation. (Contributed by RP, 8-May-2020.) |
Ref | Expression |
---|---|
trclfvcotrg | ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trclfvcotr 14371 | . 2 ⊢ (𝑅 ∈ V → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) | |
2 | fvprc 6665 | . . 3 ⊢ (¬ 𝑅 ∈ V → (t+‘𝑅) = ∅) | |
3 | 0trrel 14343 | . . . . 5 ⊢ (∅ ∘ ∅) ⊆ ∅ | |
4 | 3 | a1i 11 | . . . 4 ⊢ ((t+‘𝑅) = ∅ → (∅ ∘ ∅) ⊆ ∅) |
5 | id 22 | . . . . 5 ⊢ ((t+‘𝑅) = ∅ → (t+‘𝑅) = ∅) | |
6 | 5, 5 | coeq12d 5737 | . . . 4 ⊢ ((t+‘𝑅) = ∅ → ((t+‘𝑅) ∘ (t+‘𝑅)) = (∅ ∘ ∅)) |
7 | 4, 6, 5 | 3sstr4d 4016 | . . 3 ⊢ ((t+‘𝑅) = ∅ → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) |
8 | 2, 7 | syl 17 | . 2 ⊢ (¬ 𝑅 ∈ V → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) |
9 | 1, 8 | pm2.61i 184 | 1 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 ∅c0 4293 ∘ ccom 5561 ‘cfv 6357 t+ctcl 14347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-iota 6316 df-fun 6359 df-fv 6365 df-trcl 14349 |
This theorem is referenced by: cotrcltrcl 40077 brtrclfv2 40079 frege96d 40101 frege97d 40104 frege98d 40105 frege109d 40109 frege131d 40116 |
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