Theorem dftrrels2 38828

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 5633 (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 38670 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 38826	. 2 ⊢ TrRels = ( Trs ∩ Rels )
2		df-trs 38825	. 2 ⊢ Trs = {𝑟 ∣ ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∘ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3		inex1g 5264	. . . . 5 ⊢ (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
4	3	elv 3445	. . . 4 ⊢ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V
5		brssr 38750	. . . 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 38626	. . . . . . 7 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
8	7	elv 3445	. . . . . 6 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
9	8	biimpi 216	. . . . 5 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
10	9, 9	coeq12d 5813	. . . 4 ⊢ (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∘ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) = (𝑟 ∘ 𝑟))
11	10, 9	sseq12d 3967	. . 3 ⊢ (𝑟 ∈ Rels → (((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∘ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ (𝑟 ∘ 𝑟) ⊆ 𝑟))
12	6, 11	bitrid 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 38447	1 ⊢ TrRels = {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟}

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901   class class class wbr 5098   × cxp 5622  dom cdm 5624  ran crn 5625   ∘ ccom 5628   Rels crels 38381   S cssr 38382   Trs ctrs 38392   TrRels ctrrels 38393

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-rels 38621  df-ssr 38747  df-trs 38825  df-trrels 38826

This theorem is referenced by:  dftrrels3  38829  eltrrels2  38832  dfeqvrels2  38841