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 ↾ 𝐴 ) ) Part 𝐴 ↔ ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ErALTV 𝐴 )

Proof

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