Definition df-petparts 40091

Description: Define the class of partition-side general partition-equivalence spans.

⟨𝑟, 𝑛⟩ ∈ PetParts means:

(1) 𝑟 is a set-relation (𝑟 ∈ Rels), and

(2) 𝑛 is a membership block-carrier (𝑛 ∈ MembParts), and

(3) the block-lift span (𝑟 ⋉ (^◡ E ↾ 𝑛)) is a generalized partition on its natural quotient-carrier 𝑛 (i.e. (𝑟 ⋉ (^◡ E ↾ 𝑛)) Parts 𝑛).

This is the horizontal feasibility base object on the partition side, expressed in the type-safe Parts language.

The explicit typing (𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) is included at the definition level so later modular refinements can treat typedness as a first-class component (e.g. intersecting a typedness module with disjointness and equilibrium modules) without repeatedly restating it. In particular, it lets decompositions such as dfpetparts2 40095 be written as clean intersections whose first conjunct is exactly the typedness module ( Rels × MembParts ). (Contributed by Peter Mazsa, 19-Feb-2026.) (Revised by Peter Mazsa, 25-Feb-2026.)

Assertion

Ref	Expression
df-petparts	⊢ PetParts = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}

Distinct variable group:   𝑛,𝑟

Detailed syntax breakdown of Definition df-petparts

Step	Hyp	Ref	Expression
1		cpetparts 39350	. 2 class PetParts
2		vr	. . . . . . 7 setvar 𝑟
3	2	cv 1569	. . . . . 6 class 𝑟
4		crels 39308	. . . . . 6 class Rels
5	3, 4	wcel 2147	. . . . 5 wff 𝑟 ∈ Rels
6		vn	. . . . . . 7 setvar 𝑛
7	6	cv 1569	. . . . . 6 class 𝑛
8		cmembparts 39348	. . . . . 6 class MembParts
9	7, 8	wcel 2147	. . . . 5 wff 𝑛 ∈ MembParts
10	5, 9	wa 401	. . . 4 wff (𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts )
11		cep 5554	. . . . . . . 8 class E
12	11	ccnv 5654	. . . . . . 7 class ◡ E
13	12, 7	cres 5657	. . . . . 6 class (◡ E ↾ 𝑛)
14	3, 13	cxrn 39297	. . . . 5 class (𝑟 ⋉ (◡ E ↾ 𝑛))
15		cparts 39346	. . . . 5 class Parts
16	14, 7, 15	wbr 5106	. . . 4 wff (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛
17	10, 16	wa 401	. . 3 wff ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)
18	17, 2, 6	copab 5170	. 2 class {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}
19	1, 18	wceq 1570	1 wff PetParts = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}

This definition is used by:  dfpetparts2  40095  petseq  40099