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 ∈ 𝐴 → 𝐶 ∈ Cat )

Proof

Step	Hyp	Ref	Expression
1		arwrcl.a	⊢ 𝐴 = ( Arrow ‘ 𝐶 )
2		df-arw	⊢ Arrow = ( 𝑐 ∈ Cat ↦ ∪ ran ( Homa ‘ 𝑐 ) )
3	2	dmmptss	⊢ dom Arrow ⊆ Cat
4		elfvdm	⊢ ( 𝐹 ∈ ( Arrow ‘ 𝐶 ) → 𝐶 ∈ dom Arrow )
5	4 1	eleq2s	⊢ ( 𝐹 ∈ 𝐴 → 𝐶 ∈ dom Arrow )
6	3 5	sseldi	⊢ ( 𝐹 ∈ 𝐴 → 𝐶 ∈ Cat )