Theorem conrel2d

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

	Ref	Expression
Hypothesis	conrel1d.a	$$⊢\phi \to A^{-1}=\varnothing$$
Assertion	conrel2d	$$⊢\phi \to B\circ A=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		conrel1d.a	$$⊢\phi \to A^{-1}=\varnothing$$
2		df-rn	$$⊢\mathrm{ran}A=\mathrm{dom}{A}^{-1}$$
3	2	ineq2i	$$⊢\mathrm{dom}B\cap \mathrm{ran}A=\mathrm{dom}B\cap \mathrm{dom}{A}^{-1}$$
4	3	a1i	$$⊢\phi \to \mathrm{dom}B\cap \mathrm{ran}A=\mathrm{dom}B\cap \mathrm{dom}{A}^{-1}$$
5	1	dmeqd	$$⊢\phi \to \mathrm{dom}{A}^{-1}=\mathrm{dom}\varnothing$$
6	5	ineq2d	$$⊢\phi \to \mathrm{dom}B\cap \mathrm{dom}{A}^{-1}=\mathrm{dom}B\cap \mathrm{dom}\varnothing$$
7		dm0	$$⊢\mathrm{dom}\varnothing =\varnothing$$
8	7	ineq2i	$$⊢\mathrm{dom}B\cap \mathrm{dom}\varnothing =\mathrm{dom}B\cap \varnothing$$
9		in0	$$⊢\mathrm{dom}B\cap \varnothing =\varnothing$$
10	8 9	eqtri	$$⊢\mathrm{dom}B\cap \mathrm{dom}\varnothing =\varnothing$$
11	10	a1i	$$⊢\phi \to \mathrm{dom}B\cap \mathrm{dom}\varnothing =\varnothing$$
12	4 6 11	3eqtrd	$$⊢\phi \to \mathrm{dom}B\cap \mathrm{ran}A=\varnothing$$
13	12	coemptyd	$$⊢\phi \to B\circ A=\varnothing$$