Definition df-peters 39568

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 39575 (using typesafepets 39574 and mpets 39555). The explicit typing (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) is included for the same reason as in df-petparts 39567: 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 38809	. 2 class PetErs
2		vr	. . . . . . 7 setvar 𝑟
3	2	cv 1567	. . . . . 6 class 𝑟
4		crels 38784	. . . . . 6 class Rels
5	3, 4	wcel 2150	. . . . 5 wff 𝑟 ∈ Rels
6		vn	. . . . . . 7 setvar 𝑛
7	6	cv 1567	. . . . . 6 class 𝑛
8		ccomembers 38811	. . . . . 6 class CoMembErs
9	7, 8	wcel 2150	. . . . 5 wff 𝑛 ∈ CoMembErs
10	5, 9	wa 400	. . . 4 wff (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs )
11		cep 5564	. . . . . . . . 9 class E
12	11	ccnv 5664	. . . . . . . 8 class ◡ E
13	12, 7	cres 5667	. . . . . . 7 class (◡ E ↾ 𝑛)
14	3, 13	cxrn 38773	. . . . . 6 class (𝑟 ⋉ (◡ E ↾ 𝑛))
15	14	ccoss 38782	. . . . 5 class ≀ (𝑟 ⋉ (◡ E ↾ 𝑛))
16		cers 38807	. . . . 5 class Ers
17	15, 7, 16	wbr 5114	. . . 4 wff ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛
18	10, 17	wa 400	. . 3 wff ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)
19	18, 2, 6	copab 5178	. 2 class {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}
20	1, 19	wceq 1568	1 wff PetErs = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (◡ E ↾ 𝑛)) Ers 𝑛)}

This definition is referenced by:  dfpeters2  39573  petseq  39575