Theorem arwrcl

Description: The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017)

	Ref	Expression
Hypothesis	arwrcl.a	|- A = ( Arrow ` C )
Assertion	arwrcl	|- ( F e. A -> C e. Cat )

Proof

Step	Hyp	Ref	Expression
1		arwrcl.a	|- A = ( Arrow ` C )
2		df-arw	|- Arrow = ( c e. Cat |-> U. ran ( HomA ` c ) )
3	2	dmmptss	|- dom Arrow C_ Cat
4		elfvdm	|- ( F e. ( Arrow ` C ) -> C e. dom Arrow )
5	4 1	eleq2s	|- ( F e. A -> C e. dom Arrow )
6	3 5	sseldi	|- ( F e. A -> C e. Cat )