Definition df-peters 40092

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

⟨𝑟, 𝑛⟩ ∈ PetErs means:

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

(2) 𝑛 is a carrier recognized on the equivalence side of membership (𝑛 ∈ CoMembErs), and

(3) the coset relation of the lifted span, ≀ (𝑟 ⋉ (^◡ E ↾ 𝑛)), is an equivalence relation on its natural quotient with carrier 𝑛 (i.e. ≀ (𝑟 ⋉ (^◡ E ↾ 𝑛)) Ers 𝑛).

This packages the equivalence-view of the same lifted construction that underlies PetParts. It is designed to be parallel to PetParts so later proofs can freely choose the partition side (Parts) or the equivalence side (Ers) without rebuilding the bridge each time; the identification is provided by petseq 40099 (using typesafepets 40098 and mpets 40079). The explicit typing (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) is included for the same reason as in df-petparts 40091: to make typedness a reusable module. (Contributed by Peter Mazsa, 19-Feb-2026.) (Revised by Peter Mazsa, 25-Feb-2026.)

Assertion

Ref	Expression
df-peters	⊢ PetErs = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}

Distinct variable group:   𝑛,𝑟

Detailed syntax breakdown of Definition df-peters

Step	Hyp	Ref	Expression
1		cpeters 39333	. 2 class PetErs
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		ccomembers 39335	. . . . . 6 class CoMembErs
9	7, 8	wcel 2147	. . . . 5 wff 𝑛 ∈ CoMembErs
10	5, 9	wa 401	. . . 4 wff (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs )
11		cep 5554	. . . . . . . . 9 class E
12	11	ccnv 5654	. . . . . . . 8 class ◡ E
13	12, 7	cres 5657	. . . . . . 7 class (◡ E ↾ 𝑛)
14	3, 13	cxrn 39297	. . . . . 6 class (𝑟 ⋉ (◡ E ↾ 𝑛))
15	14	ccoss 39306	. . . . 5 class ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))
16		cers 39331	. . . . 5 class Ers
17	15, 7, 16	wbr 5106	. . . 4 wff ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛
18	10, 17	wa 401	. . 3 wff ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)
19	18, 2, 6	copab 5170	. 2 class {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}
20	1, 19	wceq 1570	1 wff PetErs = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}

This definition is used by:  dfpeters2  40097  petseq  40099