Theorem petincnvepres2

Description: A partition-equivalence theorem with intersection and general R . (Contributed by Peter Mazsa, 31-Dec-2021)

	Ref	Expression
Assertion	petincnvepres2	⊢ ( ( 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		eqvrelqseqdisj4	⊢ ( ( 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 ) )