Definition df-peters 39310

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 39317 (using typesafepets 39316 and mpets 39297). The explicit typing (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) is included for the same reason as in df-petparts 39309: 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 38551	. 2 class PetErs
2		vr	. . . . . . 7 setvar 𝑟
3	2	cv 1541	. . . . . 6 class 𝑟
4		crels 38526	. . . . . 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 38553	. . . . . 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 5525	. . . . . . . . 9 class E
12	11	ccnv 5625	. . . . . . . 8 class ◡ E
13	12, 7	cres 5628	. . . . . . 7 class (◡ E ↾ 𝑛)
14	3, 13	cxrn 38515	. . . . . 6 class (𝑟 ⋉ (◡ E ↾ 𝑛))
15	14	ccoss 38524	. . . . 5 class ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))
16		cers 38549	. . . . 5 class Ers
17	15, 7, 16	wbr 5086	. . . 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 5148	. 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  39315  petseq  39317