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Theorem dftrrels2 38556
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 (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝐴𝑢𝑢𝐵𝑦)} as opposed to the present definition of composition df-co 5697 (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝐵𝑢𝑢𝐴𝑦)} 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)) ↔ ∀𝑥𝐴𝑦𝐵 𝑧𝐶((𝑥𝑅𝑦𝑦𝑆𝑧) → 𝑥𝑇𝑧)) while the present definition of composition disarranges them: trsinxp (((𝑆 ∩ (𝐵 × 𝐶)) ∘ (𝑅 ∩ (𝐴 × 𝐵))) ⊆ (𝑇 ∩ (𝐴 × 𝐶 )) ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶((𝑥𝑅𝑦𝑦𝑆𝑧) → 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 38392 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.)

Assertion
Ref Expression
dftrrels2 TrRels = {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟}

Proof of Theorem dftrrels2
StepHypRef Expression
1 df-trrels 38554 . 2 TrRels = ( Trs ∩ Rels )
2 df-trs 38553 . 2 Trs = {𝑟 ∣ ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∘ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3 inex1g 5324 . . . . 5 (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
43elv 3482 . . . 4 (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V
5 brssr 38482 . . . 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))))
64, 5ax-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 38471 . . . . . . 7 (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
87elv 3482 . . . . . 6 (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
98biimpi 216 . . . . 5 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
109, 9coeq12d 5877 . . . 4 (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∘ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) = (𝑟𝑟))
1110, 9sseq12d 4028 . . 3 (𝑟 ∈ Rels → (((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∘ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ (𝑟𝑟) ⊆ 𝑟))
126, 11bitrid 283 . 2 (𝑟 ∈ Rels → (((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∘ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ (𝑟𝑟) ⊆ 𝑟))
131, 2, 12abeqinbi 38234 1 TrRels = {𝑟 ∈ Rels ∣ (𝑟𝑟) ⊆ 𝑟}
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1536  wcel 2105  {crab 3432  Vcvv 3477  cin 3961  wss 3962   class class class wbr 5147   × cxp 5686  dom cdm 5688  ran crn 5689  ccom 5692   Rels crels 38163   S cssr 38164   Trs ctrs 38174   TrRels ctrrels 38175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-rels 38466  df-ssr 38479  df-trs 38553  df-trrels 38554
This theorem is referenced by:  dftrrels3  38557  eltrrels2  38560  dfeqvrels2  38569
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