Definition df-peters 39139

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 39146 (using typesafepets 39145 and mpets 39126). The explicit typing (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) is included for the same reason as in df-petparts 39138: 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 38380	. 2 class PetErs
2		vr	. . . . . . 7 setvar 𝑟
3	2	cv 1541	. . . . . 6 class 𝑟
4		crels 38355	. . . . . 6 class Rels
5	3, 4	wcel 2114	. . . . 5 wff 𝑟 ∈ Rels
6		vn	. . . . . . 7 setvar 𝑛
7	6	cv 1541	. . . . . 6 class 𝑛
8		ccomembers 38382	. . . . . 6 class CoMembErs
9	7, 8	wcel 2114	. . . . 5 wff 𝑛 ∈ CoMembErs
10	5, 9	wa 395	. . . 4 wff (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs )
11		cep 5522	. . . . . . . . 9 class E
12	11	ccnv 5622	. . . . . . . 8 class ◡ E
13	12, 7	cres 5625	. . . . . . 7 class (◡ E ↾ 𝑛)
14	3, 13	cxrn 38344	. . . . . 6 class (𝑟 ⋉ (◡ E ↾ 𝑛))
15	14	ccoss 38353	. . . . 5 class ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))
16		cers 38378	. . . . 5 class Ers
17	15, 7, 16	wbr 5097	. . . 4 wff ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛
18	10, 17	wa 395	. . 3 wff ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)
19	18, 2, 6	copab 5159	. 2 class {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}
20	1, 19	wceq 1542	1 wff PetErs = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}

This definition is referenced by:  dfpeters2  39144  petseq  39146