Definition df-peters 39704

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 39711 (using typesafepets 39710 and mpets 39691). The explicit typing (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) is included for the same reason as in df-petparts 39703: 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 38945	. 2 class PetErs
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		ccomembers 38947	. . . . . 6 class CoMembErs
9	7, 8	wcel 2145	. . . . 5 wff 𝑛 ∈ CoMembErs
10	5, 9	wa 401	. . . 4 wff (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs )
11		cep 5558	. . . . . . . . 9 class E
12	11	ccnv 5658	. . . . . . . 8 class ◡ E
13	12, 7	cres 5661	. . . . . . 7 class (◡ E ↾ 𝑛)
14	3, 13	cxrn 38909	. . . . . 6 class (𝑟 ⋉ (◡ E ↾ 𝑛))
15	14	ccoss 38918	. . . . 5 class ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))
16		cers 38943	. . . . 5 class Ers
17	15, 7, 16	wbr 5107	. . . 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 5171	. 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  39709  petseq  39711