Definition df-petparts 39309

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 39313 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 38568	. 2 class PetParts
2		vr	. . . . . . 7 setvar 𝑟
3	2	cv 1541	. . . . . 6 class 𝑟
4		crels 38526	. . . . . 6 class Rels
5	3, 4	wcel 2114	. . . . 5 wff 𝑟 ∈ Rels
6		vn	. . . . . . 7 setvar 𝑛
7	6	cv 1541	. . . . . 6 class 𝑛
8		cmembparts 38566	. . . . . 6 class MembParts
9	7, 8	wcel 2114	. . . . 5 wff 𝑛 ∈ MembParts
10	5, 9	wa 395	. . . 4 wff (𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts )
11		cep 5525	. . . . . . . 8 class E
12	11	ccnv 5625	. . . . . . 7 class ◡ E
13	12, 7	cres 5628	. . . . . 6 class (◡ E ↾ 𝑛)
14	3, 13	cxrn 38515	. . . . 5 class (𝑟 ⋉ (◡ E ↾ 𝑛))
15		cparts 38564	. . . . 5 class Parts
16	14, 7, 15	wbr 5086	. . . 4 wff (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛
17	10, 16	wa 395	. . 3 wff ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)
18	17, 2, 6	copab 5148	. 2 class {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}
19	1, 18	wceq 1542	1 wff PetParts = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}

This definition is referenced by:  dfpetparts2  39313  petseq  39317