Metamath Proof Explorer

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

Ref

Expression

Assertion

petincnvepres2

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

Proof

Step	Hyp	Ref	Expression
1		eqvrelqseqdisj4	\|- ( ( EqvRel ,~ ( R i^i ( `' _E \|` A ) ) /\ ( dom ,~ ( R i^i ( `' _E \|` A ) ) /. ,~ ( R i^i ( `' _E \|` A ) ) ) = A ) -> Disj ( R i^i ( `' _E \|` A ) ) )
2	1	petlem	\|- ( ( Disj ( R i^i ( `' _E \|` A ) ) /\ ( dom ( R i^i ( `' _E \|` A ) ) /. ( R i^i ( `' _E \|` A ) ) ) = A ) <-> ( EqvRel ,~ ( R i^i ( `' _E \|` A ) ) /\ ( dom ,~ ( R i^i ( `' _E \|` A ) ) /. ,~ ( R i^i ( `' _E \|` A ) ) ) = A ) )

Step

Hyp

Ref

Expression

eqvrelqseqdisj4

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

petlem

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