Theorem isclintop

Description: The predicate "is a closed (internal binary) operations for a set". (Contributed by FL, 2-Nov-2009) (Revised by AV, 20-Jan-2020)

	Ref	Expression
Assertion	isclintop	|- ( M e. V -> ( .o. e. ( clIntOp ` M ) <-> .o. : ( M X. M ) --> M ) )

Proof

Step	Hyp	Ref	Expression
1		clintopval	|- ( M e. V -> ( clIntOp ` M ) = ( M ^m ( M X. M ) ) )
2	1	eleq2d	|- ( M e. V -> ( .o. e. ( clIntOp ` M ) <-> .o. e. ( M ^m ( M X. M ) ) ) )
3		sqxpexg	|- ( M e. V -> ( M X. M ) e. _V )
4		elmapg	|- ( ( M e. V /\ ( M X. M ) e. _V ) -> ( .o. e. ( M ^m ( M X. M ) ) <-> .o. : ( M X. M ) --> M ) )
5	3 4	mpdan	|- ( M e. V -> ( .o. e. ( M ^m ( M X. M ) ) <-> .o. : ( M X. M ) --> M ) )
6	2 5	bitrd	|- ( M e. V -> ( .o. e. ( clIntOp ` M ) <-> .o. : ( M X. M ) --> M ) )