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Metamath Proof Explorer

Theorem mpet2

Description: Member Partition-Equivalence Theorem in a shorter form. Together with mpet mpet3 , mostly in its conventional cpet and cpet2 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 with general R ). (Contributed by Peter Mazsa, 24-Sep-2021)

		Ref	Expression
	Assertion	mpet2	⊢ ( ( ^◡ E ↾ 𝐴 ) Part 𝐴 ↔ ≀ ( ^◡ E ↾ 𝐴 ) ErALTV 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		mpet	⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴 )
2		df-membpart	⊢ ( MembPart 𝐴 ↔ ( ^◡ E ↾ 𝐴 ) Part 𝐴 )
3		df-comember	⊢ ( CoMembEr 𝐴 ↔ ≀ ( ^◡ E ↾ 𝐴 ) ErALTV 𝐴 )
4	1 2 3	3bitr3i	⊢ ( ( ^◡ E ↾ 𝐴 ) Part 𝐴 ↔ ≀ ( ^◡ E ↾ 𝐴 ) ErALTV 𝐴 )