Definition df-petparts 39820

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 39824 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 39079	. 2 class PetParts
2		vr	. . . . . . 7 setvar 𝑟
3	2	cv 1569	. . . . . 6 class 𝑟
4		crels 39037	. . . . . 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 39077	. . . . . 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 5546	. . . . . . . 8 class E
12	11	ccnv 5646	. . . . . . 7 class ◡ E
13	12, 7	cres 5649	. . . . . 6 class (◡ E ↾ 𝑛)
14	3, 13	cxrn 39026	. . . . 5 class (𝑟 ⋉ (◡ E ↾ 𝑛))
15		cparts 39075	. . . . 5 class Parts
16	14, 7, 15	wbr 5102	. . . 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 5166	. 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  39824  petseq  39828