Metamath Proof Explorer

Theorem pet

Description: Partition-Equivalence Theorem with general R while preserving the restricted converse epsilon relation of mpet2 (as opposed to petincnvepres ). A class is a partition by a range Cartesian product with general R and the restricted converse element class if and only if the cosets by the range Cartesian product are in an equivalence relation on it. Cf. br1cossxrncnvepres .

This theorem (together with pets and pet2 ) is the main result of my investigation into set theory. It is no more general than the conventional Member Partition-Equivalence Theorem mpet , mpet2 and mpet3 (because you cannot set R in this theorem in such a way that you get mpet2 ), i.e., it is not the hypothetical General Partition-Equivalence Theorem gpet |- ( R Part A <-> ,R ErALTV A ) , but this one has a general part that mpet2 lacks: R , which is sufficient for my future application of set theory, for my purpose outside of set theory. (Contributed by Peter Mazsa, 23-Sep-2021)

Ref Expression
Assertion pet ⊢ R ⋉ E -1 ↾ A Part A ↔ ≀ R ⋉ E -1 ↾ A ErALTV A

Proof

Step Hyp Ref Expression
1 pet2 ⊢ Disj R ⋉ E -1 ↾ A ∧ dom ⁡ R ⋉ E -1 ↾ A / R ⋉ E -1 ↾ A = A ↔ EqvRel ≀ R ⋉ E -1 ↾ A ∧ dom ⁡ ≀ R ⋉ E -1 ↾ A / ≀ R ⋉ E -1 ↾ A = A
2 dfpart2 ⊢ R ⋉ E -1 ↾ A Part A ↔ Disj R ⋉ E -1 ↾ A ∧ dom ⁡ R ⋉ E -1 ↾ A / R ⋉ E -1 ↾ A = A
3 dferALTV2 ⊢ ≀ R ⋉ E -1 ↾ A ErALTV A ↔ EqvRel ≀ R ⋉ E -1 ↾ A ∧ dom ⁡ ≀ R ⋉ E -1 ↾ A / ≀ R ⋉ E -1 ↾ A = A
4 1 2 3 3bitr4i ⊢ R ⋉ E -1 ↾ A Part A ↔ ≀ R ⋉ E -1 ↾ A ErALTV A