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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 15047 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) | |
| 2 | 1 | adantr 480 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) | 
| 3 | simpr 484 | . . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel 𝑅) | |
| 4 | relssdmrn 6287 | . . . . 5 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) | |
| 5 | ssequn1 4185 | . . . . . 6 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)) | |
| 6 | 5 | biimpi 216 | . . . . 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 4019 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (dom 𝑅 × ran 𝑅)) | 
| 9 | xpss 5700 | . . 3 ⊢ (dom 𝑅 × ran 𝑅) ⊆ (V × V) | |
| 10 | 8, 9 | sstrdi 3995 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (V × V)) | 
| 11 | df-rel 5691 | . 2 ⊢ (Rel (t+‘𝑅) ↔ (t+‘𝑅) ⊆ (V × V)) | |
| 12 | 10, 11 | sylibr 234 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∪ cun 3948 ⊆ wss 3950 × cxp 5682 dom cdm 5684 ran crn 5685 Rel wrel 5689 ‘cfv 6560 t+ctcl 15025 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-iota 6513 df-fun 6562 df-fv 6568 df-trcl 15027 | 
| This theorem is referenced by: frege124d 43779 frege129d 43781 frege133d 43783 | 
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