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 \|X. ( `' _E \|` A ) ) /\ ( dom ( R \|X. ( `' _E \|` A ) ) /. ( R \|X. ( `' _E \|` A ) ) ) = A ) <-> ( EqvRel ,~ ( R \|X. ( `' _E \|` A ) ) /\ ( dom ,~ ( R \|X. ( `' _E \|` A ) ) /. ,~ ( R \|X. ( `' _E \|` A ) ) ) = A ) )

Ref

Expression

Assertion

pet2

|- ( ( Disj ( R |X. ( `' _E |` A ) ) /\ ( dom ( R |X. ( `' _E |` A ) ) /. ( R |X. ( `' _E |` A ) ) ) = A ) <-> ( EqvRel ,~ ( R |X. ( `' _E |` A ) ) /\ ( dom ,~ ( R |X. ( `' _E |` A ) ) /. ,~ ( R |X. ( `' _E |` A ) ) ) = A ) )

Proof

Step	Hyp	Ref	Expression
1		eqvrelqseqdisj5	\|- ( ( EqvRel ,~ ( R \|X. ( `' _E \|` A ) ) /\ ( dom ,~ ( R \|X. ( `' _E \|` A ) ) /. ,~ ( R \|X. ( `' _E \|` A ) ) ) = A ) -> Disj ( R \|X. ( `' _E \|` A ) ) )
2	1	petlem	\|- ( ( Disj ( R \|X. ( `' _E \|` A ) ) /\ ( dom ( R \|X. ( `' _E \|` A ) ) /. ( R \|X. ( `' _E \|` A ) ) ) = A ) <-> ( EqvRel ,~ ( R \|X. ( `' _E \|` A ) ) /\ ( dom ,~ ( R \|X. ( `' _E \|` A ) ) /. ,~ ( R \|X. ( `' _E \|` A ) ) ) = A ) )

Step

Hyp

Ref

Expression

eqvrelqseqdisj5

 |-  ( ( EqvRel ,~ ( R |X. ( `' _E |` A ) ) /\ ( dom ,~ ( R |X. ( `' _E |` A ) ) /. ,~ ( R |X. ( `' _E |` A ) ) ) = A ) -> Disj ( R |X. ( `' _E |` A ) ) )

petlem

 |-  ( ( Disj ( R |X. ( `' _E |` A ) ) /\ ( dom ( R |X. ( `' _E |` A ) ) /. ( R |X. ( `' _E |` A ) ) ) = A ) <-> ( EqvRel ,~ ( R |X. ( `' _E |` A ) ) /\ ( dom ,~ ( R |X. ( `' _E |` A ) ) /. ,~ ( R |X. ( `' _E |` A ) ) ) = A ) )