Definition df-petparts 39472

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 39476 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 38731	. 2 class PetParts
2		vr	. . . . . . 7 setvar 𝑟
3	2	cv 1561	. . . . . 6 class 𝑟
4		crels 38689	. . . . . 6 class Rels
5	3, 4	wcel 2144	. . . . 5 wff 𝑟 ∈ Rels
6		vn	. . . . . . 7 setvar 𝑛
7	6	cv 1561	. . . . . 6 class 𝑛
8		cmembparts 38729	. . . . . 6 class MembParts
9	7, 8	wcel 2144	. . . . 5 wff 𝑛 ∈ MembParts
10	5, 9	wa 399	. . . 4 wff (𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts )
11		cep 5548	. . . . . . . 8 class E
12	11	ccnv 5648	. . . . . . 7 class ◡ E
13	12, 7	cres 5651	. . . . . 6 class (◡ E ↾ 𝑛)
14	3, 13	cxrn 38678	. . . . 5 class (𝑟 ⋉ (◡ E ↾ 𝑛))
15		cparts 38727	. . . . 5 class Parts
16	14, 7, 15	wbr 5102	. . . 4 wff (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛
17	10, 16	wa 399	. . 3 wff ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)
18	17, 2, 6	copab 5164	. 2 class {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}
19	1, 18	wceq 1562	1 wff PetParts = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (◡ E ↾ 𝑛)) Parts 𝑛)}

This definition is referenced by:  dfpetparts2  39476  petseq  39480