Theorem pets

Description: Partition-Equivalence Theorem with general R , with binary relations. This theorem (together with pet and pet2 ) is the main result of my investigation into set theory, cf. the comment of pet . (Contributed by Peter Mazsa, 23-Sep-2021)

	Ref	Expression
Assertion	pets	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) Parts 𝐴 ↔ ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) Ers 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		pet	⊢ ( ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) Part 𝐴 ↔ ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ErALTV 𝐴 )
2		xrncnvepresex	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ∈ V )
3		brpartspart	⊢ ( ( 𝐴 ∈ 𝑉 ∧ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ∈ V ) → ( ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) Parts 𝐴 ↔ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) Part 𝐴 ) )
4	2 3	syldan	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) Parts 𝐴 ↔ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) Part 𝐴 ) )
5		1cossxrncnvepresex	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ∈ V )
6		brerser	⊢ ( ( 𝐴 ∈ 𝑉 ∧ ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ∈ V ) → ( ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) Ers 𝐴 ↔ ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ErALTV 𝐴 ) )
7	5 6	syldan	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) Ers 𝐴 ↔ ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ErALTV 𝐴 ) )
8	4 7	bibi12d	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( ( ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) Parts 𝐴 ↔ ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) Ers 𝐴 ) ↔ ( ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) Part 𝐴 ↔ ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ErALTV 𝐴 ) ) )
9	1 8	mpbiri	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) Parts 𝐴 ↔ ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) Ers 𝐴 ) )