Theorem elcnvintab

Description: Two ways of saying a set is an element of the converse of the intersection of a class. (Contributed by RP, 19-Aug-2020)

	Ref	Expression
Assertion	elcnvintab	|- ( A e. `' |^| { x | ph } <-> ( A e. ( _V X. _V ) /\ A. x ( ph -> A e. `' x ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	|- ( y e. ( _V X. _V ) |-> <. ( 2nd ` y ) , ( 1st ` y ) >. ) = ( y e. ( _V X. _V ) |-> <. ( 2nd ` y ) , ( 1st ` y ) >. )
2	1	elcnvlem	|- ( A e. `' |^| { x | ph } <-> ( A e. ( _V X. _V ) /\ ( ( y e. ( _V X. _V ) |-> <. ( 2nd ` y ) , ( 1st ` y ) >. ) ` A ) e. |^| { x | ph } ) )
3	1	elcnvlem	|- ( A e. `' x <-> ( A e. ( _V X. _V ) /\ ( ( y e. ( _V X. _V ) |-> <. ( 2nd ` y ) , ( 1st ` y ) >. ) ` A ) e. x ) )
4	2 3	elmapintab	|- ( A e. `' |^| { x | ph } <-> ( A e. ( _V X. _V ) /\ A. x ( ph -> A e. `' x ) ) )