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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 14974 | . 2 ⊢ (𝑅 ∈ V → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) | |
2 | fvprc 6883 | . . 3 ⊢ (¬ 𝑅 ∈ V → (t+‘𝑅) = ∅) | |
3 | 0trrel 14946 | . . . . 5 ⊢ (∅ ∘ ∅) ⊆ ∅ | |
4 | 3 | a1i 11 | . . . 4 ⊢ ((t+‘𝑅) = ∅ → (∅ ∘ ∅) ⊆ ∅) |
5 | id 22 | . . . . 5 ⊢ ((t+‘𝑅) = ∅ → (t+‘𝑅) = ∅) | |
6 | 5, 5 | coeq12d 5861 | . . . 4 ⊢ ((t+‘𝑅) = ∅ → ((t+‘𝑅) ∘ (t+‘𝑅)) = (∅ ∘ ∅)) |
7 | 4, 6, 5 | 3sstr4d 4025 | . . 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 1534 ∈ wcel 2099 Vcvv 3469 ⊆ wss 3944 ∅c0 4318 ∘ ccom 5676 ‘cfv 6542 t+ctcl 14950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-iota 6494 df-fun 6544 df-fv 6550 df-trcl 14952 |
This theorem is referenced by: cotrcltrcl 43068 brtrclfv2 43070 frege96d 43092 frege97d 43095 frege98d 43096 frege109d 43100 frege131d 43107 |
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