Theorem elhomai

Description: Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017)

	Ref	Expression
Hypotheses	homarcl.h	⊢ 𝐻 = ( Homa ‘ 𝐶 )
	homafval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	homafval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	homaval.j	⊢ 𝐽 = ( Hom ‘ 𝐶 )
	homaval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	homaval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	elhomai.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐽 𝑌 ) )
Assertion	elhomai	⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ( 𝑋 𝐻 𝑌 ) 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		homarcl.h	⊢ 𝐻 = ( Homa ‘ 𝐶 )
2		homafval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		homafval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
4		homaval.j	⊢ 𝐽 = ( Hom ‘ 𝐶 )
5		homaval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		homaval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		elhomai.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐽 𝑌 ) )
8		eqidd	⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ = ⟨ 𝑋 , 𝑌 ⟩ )
9	1 2 3 4 5 6	elhoma	⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ ( 𝑋 𝐻 𝑌 ) 𝐹 ↔ ( ⟨ 𝑋 , 𝑌 ⟩ = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝐹 ∈ ( 𝑋 𝐽 𝑌 ) ) ) )
10	8 7 9	mpbir2and	⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ( 𝑋 𝐻 𝑌 ) 𝐹 )