Theorem elfvov1

Description: Utility theorem: reverse closure for any operation that results in a function. (Contributed by SN, 18-May-2025)

	Ref	Expression
Hypotheses	elfvov1.o	|- Rel dom O
	elfvov1.s	|- S = ( I O R )
	elfvov1.x	|- ( ph -> X e. ( S ` Y ) )
Assertion	elfvov1	|- ( ph -> I e. _V )

Proof

Step	Hyp	Ref	Expression
1		elfvov1.o	|- Rel dom O
2		elfvov1.s	|- S = ( I O R )
3		elfvov1.x	|- ( ph -> X e. ( S ` Y ) )
4		n0i	|- ( X e. ( S ` Y ) -> -. ( S ` Y ) = (/) )
5	3 4	syl	|- ( ph -> -. ( S ` Y ) = (/) )
6	1	ovprc1	|- ( -. I e. _V -> ( I O R ) = (/) )
7	2 6	eqtrid	|- ( -. I e. _V -> S = (/) )
8	7	fveq1d	|- ( -. I e. _V -> ( S ` Y ) = ( (/) ` Y ) )
9		0fv	|- ( (/) ` Y ) = (/)
10	8 9	eqtrdi	|- ( -. I e. _V -> ( S ` Y ) = (/) )
11	5 10	nsyl2	|- ( ph -> I e. _V )