Theorem arwdmcd

Description: Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017)

	Ref	Expression
Hypothesis	arwrcl.a	⊢ 𝐴 = ( Arrow ‘ 𝐶 )
Assertion	arwdmcd	⊢ ( 𝐹 ∈ 𝐴 → 𝐹 = ⟨ ( doma ‘ 𝐹 ) , ( coda ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ )

Proof

Step	Hyp	Ref	Expression
1		arwrcl.a	⊢ 𝐴 = ( Arrow ‘ 𝐶 )
2		eqid	⊢ ( Homa ‘ 𝐶 ) = ( Homa ‘ 𝐶 )
3	1 2	arwhoma	⊢ ( 𝐹 ∈ 𝐴 → 𝐹 ∈ ( ( doma ‘ 𝐹 ) ( Homa ‘ 𝐶 ) ( coda ‘ 𝐹 ) ) )
4	2	homadmcd	⊢ ( 𝐹 ∈ ( ( doma ‘ 𝐹 ) ( Homa ‘ 𝐶 ) ( coda ‘ 𝐹 ) ) → 𝐹 = ⟨ ( doma ‘ 𝐹 ) , ( coda ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ )
5	3 4	syl	⊢ ( 𝐹 ∈ 𝐴 → 𝐹 = ⟨ ( doma ‘ 𝐹 ) , ( coda ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ )