Theorem dmnonrel

Description: The domain of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020)

	Ref	Expression
Assertion	dmnonrel	⊢ dom ( 𝐴 ∖ ◡ ◡ 𝐴 ) = ∅

Proof

Step	Hyp	Ref	Expression
1		dfdm4	⊢ dom ( 𝐴 ∖ ◡ ◡ 𝐴 ) = ran ◡ ( 𝐴 ∖ ◡ ◡ 𝐴 )
2		cnvnonrel	⊢ ◡ ( 𝐴 ∖ ◡ ◡ 𝐴 ) = ∅
3	2	rneqi	⊢ ran ◡ ( 𝐴 ∖ ◡ ◡ 𝐴 ) = ran ∅
4		rn0	⊢ ran ∅ = ∅
5	1 3 4	3eqtri	⊢ dom ( 𝐴 ∖ ◡ ◡ 𝐴 ) = ∅