This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )