Definition df-petparts 39703

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 39707 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 38962	. 2 class PetParts
2		vr	. . . . . . 7 setvar 𝑟
3	2	cv 1569	. . . . . 6 class 𝑟
4		crels 38920	. . . . . 6 class Rels
5	3, 4	wcel 2145	. . . . 5 wff 𝑟 ∈ Rels
6		vn	. . . . . . 7 setvar 𝑛
7	6	cv 1569	. . . . . 6 class 𝑛
8		cmembparts 38960	. . . . . 6 class MembParts
9	7, 8	wcel 2145	. . . . 5 wff 𝑛 ∈ MembParts
10	5, 9	wa 401	. . . 4 wff (𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts )
11		cep 5558	. . . . . . . 8 class E
12	11	ccnv 5658	. . . . . . 7 class ◡ E
13	12, 7	cres 5661	. . . . . 6 class (◡ E ↾ 𝑛)
14	3, 13	cxrn 38909	. . . . 5 class (𝑟 ⋉ (◡ E ↾ 𝑛))
15		cparts 38958	. . . . 5 class Parts
16	14, 7, 15	wbr 5107	. . . 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 5171	. 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  39707  petseq  39711