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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 14955 | . 2 ⊢ (𝑅 ∈ V → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) | |
2 | fvprc 6883 | . . 3 ⊢ (¬ 𝑅 ∈ V → (t+‘𝑅) = ∅) | |
3 | 0trrel 14927 | . . . . 5 ⊢ (∅ ∘ ∅) ⊆ ∅ | |
4 | 3 | a1i 11 | . . . 4 ⊢ ((t+‘𝑅) = ∅ → (∅ ∘ ∅) ⊆ ∅) |
5 | id 22 | . . . . 5 ⊢ ((t+‘𝑅) = ∅ → (t+‘𝑅) = ∅) | |
6 | 5, 5 | coeq12d 5864 | . . . 4 ⊢ ((t+‘𝑅) = ∅ → ((t+‘𝑅) ∘ (t+‘𝑅)) = (∅ ∘ ∅)) |
7 | 4, 6, 5 | 3sstr4d 4029 | . . 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 182 | 1 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3948 ∅c0 4322 ∘ ccom 5680 ‘cfv 6543 t+ctcl 14931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-iota 6495 df-fun 6545 df-fv 6551 df-trcl 14933 |
This theorem is referenced by: cotrcltrcl 42466 brtrclfv2 42468 frege96d 42490 frege97d 42493 frege98d 42494 frege109d 42498 frege131d 42505 |
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