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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 14572 | . 2 ⊢ (𝑅 ∈ V → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) | |
2 | fvprc 6709 | . . 3 ⊢ (¬ 𝑅 ∈ V → (t+‘𝑅) = ∅) | |
3 | 0trrel 14544 | . . . . 5 ⊢ (∅ ∘ ∅) ⊆ ∅ | |
4 | 3 | a1i 11 | . . . 4 ⊢ ((t+‘𝑅) = ∅ → (∅ ∘ ∅) ⊆ ∅) |
5 | id 22 | . . . . 5 ⊢ ((t+‘𝑅) = ∅ → (t+‘𝑅) = ∅) | |
6 | 5, 5 | coeq12d 5733 | . . . 4 ⊢ ((t+‘𝑅) = ∅ → ((t+‘𝑅) ∘ (t+‘𝑅)) = (∅ ∘ ∅)) |
7 | 4, 6, 5 | 3sstr4d 3948 | . . 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 185 | 1 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ⊆ wss 3866 ∅c0 4237 ∘ ccom 5555 ‘cfv 6380 t+ctcl 14548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-int 4860 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-iota 6338 df-fun 6382 df-fv 6388 df-trcl 14550 |
This theorem is referenced by: cotrcltrcl 41010 brtrclfv2 41012 frege96d 41034 frege97d 41037 frege98d 41038 frege109d 41042 frege131d 41049 |
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