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Mirrors > Home > MPE Home > Th. List > Mathboxes > dftrrels2 | Structured version Visualization version GIF version |
Description: Alternate definition of
the class of transitive relations.
I'd prefer to define the class of transitive relations by using the definition of composition by [Suppes] p. 63. df-coSUP (𝐴 ∘ 𝐵) = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥𝐴𝑢 ∧ 𝑢𝐵𝑦)} as opposed to the present definition of composition df-co 5598 (𝐴 ∘ 𝐵) = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥𝐵𝑢 ∧ 𝑢𝐴𝑦)} because the Suppes definition keeps the order of 𝐴, 𝐵, 𝐶, 𝑅, 𝑆, 𝑇 by default in trsinxpSUP (((𝑅 ∩ (𝐴 × 𝐵)) ∘ (𝑆 ∩ (𝐵 × 𝐶))) ⊆ (𝑇 ∩ (𝐴 × 𝐶)) ↔ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵∀ 𝑧 ∈ 𝐶((𝑥𝑅𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑇𝑧)) while the present definition of composition disarranges them: trsinxp (((𝑆 ∩ (𝐵 × 𝐶)) ∘ (𝑅 ∩ (𝐴 × 𝐵))) ⊆ (𝑇 ∩ (𝐴 × 𝐶 )) ↔ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵∀𝑧 ∈ 𝐶((𝑥𝑅𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑇𝑧) ). This is not mission critical to me, the implication of the Suppes definition is just more aesthetic, at least in the above case. If we swap to the Suppes definition of class composition, I would define the present class of all transitive sets as df-trsSUP and I would consider to switch the definition of the class of cosets by 𝑅 from the present df-coss 36537 to a df-cossSUP. But perhaps there is a mathematical reason to keep the present definition of composition. (Contributed by Peter Mazsa, 21-Jul-2021.) |
Ref | Expression |
---|---|
dftrrels2 | ⊢ TrRels = {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-trrels 36687 | . 2 ⊢ TrRels = ( Trs ∩ Rels ) | |
2 | df-trs 36686 | . 2 ⊢ Trs = {𝑟 ∣ ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∘ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} | |
3 | inex1g 5243 | . . . . 5 ⊢ (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V) | |
4 | 3 | elv 3438 | . . . 4 ⊢ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V |
5 | brssr 36619 | . . . 4 ⊢ ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V → (((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∘ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∘ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)))) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ (((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∘ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∘ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) |
7 | elrels6 36608 | . . . . . . 7 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)) | |
8 | 7 | elv 3438 | . . . . . 6 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
9 | 8 | biimpi 215 | . . . . 5 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
10 | 9, 9 | coeq12d 5773 | . . . 4 ⊢ (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∘ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) = (𝑟 ∘ 𝑟)) |
11 | 10, 9 | sseq12d 3954 | . . 3 ⊢ (𝑟 ∈ Rels → (((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∘ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ (𝑟 ∘ 𝑟) ⊆ 𝑟)) |
12 | 6, 11 | syl5bb 283 | . 2 ⊢ (𝑟 ∈ Rels → (((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∘ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ (𝑟 ∘ 𝑟) ⊆ 𝑟)) |
13 | 1, 2, 12 | abeqinbi 36393 | 1 ⊢ TrRels = {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1539 ∈ wcel 2106 {crab 3068 Vcvv 3432 ∩ cin 3886 ⊆ wss 3887 class class class wbr 5074 × cxp 5587 dom cdm 5589 ran crn 5590 ∘ ccom 5593 Rels crels 36335 S cssr 36336 Trs ctrs 36346 TrRels ctrrels 36347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-rels 36603 df-ssr 36616 df-trs 36686 df-trrels 36687 |
This theorem is referenced by: dftrrels3 36690 eltrrels2 36693 dfeqvrels2 36701 |
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