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Metamath Proof Explorer

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 ) )