Theorem mpets2

Description: Member Partition-Equivalence Theorem with binary relations, cf. mpet2 . (Contributed by Peter Mazsa, 24-Sep-2021)

	Ref	Expression
Assertion	mpets2	⊢ ( 𝐴 ∈ 𝑉 → ( ( ◡ E ↾ 𝐴 ) Parts 𝐴 ↔ ≀ ( ◡ E ↾ 𝐴 ) Ers 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		mpet2	⊢ ( ( ◡ E ↾ 𝐴 ) Part 𝐴 ↔ ≀ ( ◡ E ↾ 𝐴 ) ErALTV 𝐴 )
2		cnvepresex	⊢ ( 𝐴 ∈ 𝑉 → ( ◡ E ↾ 𝐴 ) ∈ V )
3		brpartspart	⊢ ( ( 𝐴 ∈ 𝑉 ∧ ( ◡ E ↾ 𝐴 ) ∈ V ) → ( ( ◡ E ↾ 𝐴 ) Parts 𝐴 ↔ ( ◡ E ↾ 𝐴 ) Part 𝐴 ) )
4	2 3	mpdan	⊢ ( 𝐴 ∈ 𝑉 → ( ( ◡ E ↾ 𝐴 ) Parts 𝐴 ↔ ( ◡ E ↾ 𝐴 ) Part 𝐴 ) )
5		1cosscnvepresex	⊢ ( 𝐴 ∈ 𝑉 → ≀ ( ◡ E ↾ 𝐴 ) ∈ V )
6		brerser	⊢ ( ( 𝐴 ∈ 𝑉 ∧ ≀ ( ◡ E ↾ 𝐴 ) ∈ V ) → ( ≀ ( ◡ E ↾ 𝐴 ) Ers 𝐴 ↔ ≀ ( ◡ E ↾ 𝐴 ) ErALTV 𝐴 ) )
7	5 6	mpdan	⊢ ( 𝐴 ∈ 𝑉 → ( ≀ ( ◡ E ↾ 𝐴 ) Ers 𝐴 ↔ ≀ ( ◡ E ↾ 𝐴 ) ErALTV 𝐴 ) )
8	4 7	bibi12d	⊢ ( 𝐴 ∈ 𝑉 → ( ( ( ◡ E ↾ 𝐴 ) Parts 𝐴 ↔ ≀ ( ◡ E ↾ 𝐴 ) Ers 𝐴 ) ↔ ( ( ◡ E ↾ 𝐴 ) Part 𝐴 ↔ ≀ ( ◡ E ↾ 𝐴 ) ErALTV 𝐴 ) ) )
9	1 8	mpbiri	⊢ ( 𝐴 ∈ 𝑉 → ( ( ◡ E ↾ 𝐴 ) Parts 𝐴 ↔ ≀ ( ◡ E ↾ 𝐴 ) Ers 𝐴 ) )