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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 14220 | . 2 ⊢ (𝑅 ∈ V → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) | |
2 | fvprc 6486 | . . 3 ⊢ (¬ 𝑅 ∈ V → (t+‘𝑅) = ∅) | |
3 | 0trrel 14192 | . . . . 5 ⊢ (∅ ∘ ∅) ⊆ ∅ | |
4 | 3 | a1i 11 | . . . 4 ⊢ ((t+‘𝑅) = ∅ → (∅ ∘ ∅) ⊆ ∅) |
5 | id 22 | . . . . 5 ⊢ ((t+‘𝑅) = ∅ → (t+‘𝑅) = ∅) | |
6 | 5, 5 | coeq12d 5578 | . . . 4 ⊢ ((t+‘𝑅) = ∅ → ((t+‘𝑅) ∘ (t+‘𝑅)) = (∅ ∘ ∅)) |
7 | 4, 6, 5 | 3sstr4d 3900 | . . 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 177 | 1 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1507 ∈ wcel 2048 Vcvv 3409 ⊆ wss 3825 ∅c0 4173 ∘ ccom 5404 ‘cfv 6182 t+ctcl 14196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3678 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-int 4744 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-iota 6146 df-fun 6184 df-fv 6190 df-trcl 14198 |
This theorem is referenced by: cotrcltrcl 39378 brtrclfv2 39380 frege96d 39402 frege97d 39405 frege98d 39406 frege109d 39410 frege131d 39417 |
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