Theorem epihom

Description: An epimorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypotheses	isepi.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	isepi.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	isepi.o	⊢ · = ( comp ‘ 𝐶 )
	isepi.e	⊢ 𝐸 = ( Epi ‘ 𝐶 )
	isepi.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	isepi.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	isepi.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	epihom	⊢ ( 𝜑 → ( 𝑋 𝐸 𝑌 ) ⊆ ( 𝑋 𝐻 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		isepi.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
2		isepi.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
3		isepi.o	⊢ · = ( comp ‘ 𝐶 )
4		isepi.e	⊢ 𝐸 = ( Epi ‘ 𝐶 )
5		isepi.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
6		isepi.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
7		isepi.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
8	1 2 3 4 5 6 7	isepi	⊢ ( 𝜑 → ( 𝑓 ∈ ( 𝑋 𝐸 𝑌 ) ↔ ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 Fun ◡ ( 𝑔 ∈ ( 𝑌 𝐻 𝑧 ) ↦ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑧 ) 𝑓 ) ) ) ) )
9		simpl	⊢ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 Fun ◡ ( 𝑔 ∈ ( 𝑌 𝐻 𝑧 ) ↦ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑧 ) 𝑓 ) ) ) → 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) )
10	8 9	syl6bi	⊢ ( 𝜑 → ( 𝑓 ∈ ( 𝑋 𝐸 𝑌 ) → 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ) )
11	10	ssrdv	⊢ ( 𝜑 → ( 𝑋 𝐸 𝑌 ) ⊆ ( 𝑋 𝐻 𝑌 ) )