Theorem dftrrels2 36312

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 5534 (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 36160 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 ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟}

Proof of Theorem dftrrels2

Step	Hyp	Ref	Expression
1		df-trrels 36310	. 2 ⊢ TrRels = ( Trs ∩ Rels )
2		df-trs 36309	. 2 ⊢ Trs = {𝑟 ∣ ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∘ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3		inex1g 5187	. . . . 5 ⊢ (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
4	3	elv 3404	. . . 4 ⊢ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V
5		brssr 36242	. . . 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 36231	. . . . . . 7 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
8	7	elv 3404	. . . . . 6 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
9	8	biimpi 219	. . . . 5 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
10	9, 9	coeq12d 5707	. . . 4 ⊢ (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∘ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) = (𝑟 ∘ 𝑟))
11	10, 9	sseq12d 3910	. . 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 286	. 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 36016	1 ⊢ TrRels = {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟}

Syntax hints:   ↔ wb 209   = wceq 1542   ∈ wcel 2114  {crab 3057  Vcvv 3398   ∩ cin 3842   ⊆ wss 3843   class class class wbr 5030   × cxp 5523  dom cdm 5525  ran crn 5526   ∘ ccom 5529   Rels crels 35958   S cssr 35959   Trs ctrs 35969   TrRels ctrrels 35970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-rels 36226  df-ssr 36239  df-trs 36309  df-trrels 36310

This theorem is referenced by:  dftrrels3  36313  eltrrels2  36316  dfeqvrels2  36324