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Mirrors > Home > MPE Home > Th. List > reltrclfv | Structured version Visualization version GIF version |
Description: The transitive closure of a relation is a relation. (Contributed by RP, 9-May-2020.) |
Ref | Expression |
---|---|
reltrclfv | ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trclfvub 14358 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) | |
2 | 1 | adantr 484 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) |
3 | simpr 488 | . . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel 𝑅) | |
4 | relssdmrn 6088 | . . . . 5 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) | |
5 | ssequn1 4107 | . . . . . 6 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)) | |
6 | 5 | biimpi 219 | . . . . 5 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)) |
7 | 3, 4, 6 | 3syl 18 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)) |
8 | 2, 7 | sseqtrd 3955 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (dom 𝑅 × ran 𝑅)) |
9 | xpss 5535 | . . 3 ⊢ (dom 𝑅 × ran 𝑅) ⊆ (V × V) | |
10 | 8, 9 | sstrdi 3927 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (V × V)) |
11 | df-rel 5526 | . 2 ⊢ (Rel (t+‘𝑅) ↔ (t+‘𝑅) ⊆ (V × V)) | |
12 | 10, 11 | sylibr 237 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∪ cun 3879 ⊆ wss 3881 × cxp 5517 dom cdm 5519 ran crn 5520 Rel wrel 5524 ‘cfv 6324 t+ctcl 14336 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-iota 6283 df-fun 6326 df-fv 6332 df-trcl 14338 |
This theorem is referenced by: frege124d 40462 frege129d 40464 frege133d 40466 |
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