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Metamath Proof Explorer

Theorem pet2

Description: Partition-Equivalence Theorem, with general R . This theorem (together with pet and pets ) is the main result of my investigation into set theory, see the comment of pet . (Contributed by Peter Mazsa, 24-May-2021) (Revised by Peter Mazsa, 23-Sep-2021)

Ref Expression
Assertion 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 ) )

Proof

Step Hyp Ref Expression
1 eqvrelqseqdisj5 ⊢ ( ( EqvRel ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ∧ ( dom ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) / ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ) = 𝐴 ) → Disj ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) )
2 1 petlem ⊢ ( ( Disj ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ∧ ( dom ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) / ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ) = 𝐴 ) ↔ ( EqvRel ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ∧ ( dom ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) / ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ) = 𝐴 ) )