Theorem monhom

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

	Ref	Expression
Hypotheses	ismon.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	ismon.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	ismon.o	⊢ · = ( comp ‘ 𝐶 )
	ismon.s	⊢ 𝑀 = ( Mono ‘ 𝐶 )
	ismon.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	ismon.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	ismon.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	monhom	⊢ ( 𝜑 → ( 𝑋 𝑀 𝑌 ) ⊆ ( 𝑋 𝐻 𝑌 ) )

Proof

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