Theorem conrel2d

Description: Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019)

	Ref	Expression
Hypothesis	conrel1d.a	|- ( ph -> `' A = (/) )
Assertion	conrel2d	|- ( ph -> ( B o. A ) = (/) )

Proof

Step	Hyp	Ref	Expression
1		conrel1d.a	|- ( ph -> `' A = (/) )
2		df-rn	|- ran A = dom `' A
3	2	ineq2i	|- ( dom B i^i ran A ) = ( dom B i^i dom `' A )
4	3	a1i	|- ( ph -> ( dom B i^i ran A ) = ( dom B i^i dom `' A ) )
5	1	dmeqd	|- ( ph -> dom `' A = dom (/) )
6	5	ineq2d	|- ( ph -> ( dom B i^i dom `' A ) = ( dom B i^i dom (/) ) )
7		dm0	|- dom (/) = (/)
8	7	ineq2i	|- ( dom B i^i dom (/) ) = ( dom B i^i (/) )
9		in0	|- ( dom B i^i (/) ) = (/)
10	8 9	eqtri	|- ( dom B i^i dom (/) ) = (/)
11	10	a1i	|- ( ph -> ( dom B i^i dom (/) ) = (/) )
12	4 6 11	3eqtrd	|- ( ph -> ( dom B i^i ran A ) = (/) )
13	12	coemptyd	|- ( ph -> ( B o. A ) = (/) )