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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 14817 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) | |
2 | 1 | adantr 481 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) |
3 | simpr 485 | . . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel 𝑅) | |
4 | relssdmrn 6206 | . . . . 5 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) | |
5 | ssequn1 4127 | . . . . . 6 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)) | |
6 | 5 | biimpi 215 | . . . . 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 3972 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (dom 𝑅 × ran 𝑅)) |
9 | xpss 5636 | . . 3 ⊢ (dom 𝑅 × ran 𝑅) ⊆ (V × V) | |
10 | 8, 9 | sstrdi 3944 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (V × V)) |
11 | df-rel 5627 | . 2 ⊢ (Rel (t+‘𝑅) ↔ (t+‘𝑅) ⊆ (V × V)) | |
12 | 10, 11 | sylibr 233 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 Vcvv 3441 ∪ cun 3896 ⊆ wss 3898 × cxp 5618 dom cdm 5620 ran crn 5621 Rel wrel 5625 ‘cfv 6479 t+ctcl 14795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-iota 6431 df-fun 6481 df-fv 6487 df-trcl 14797 |
This theorem is referenced by: frege124d 41690 frege129d 41692 frege133d 41694 |
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