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Metamath Proof Explorer

Theorem subcss2

Description: The morphisms of a subcategory are a subset of the morphisms of the original. (Contributed by Mario Carneiro, 4-Jan-2017)

Ref Expression
Hypotheses subcss1.1 ⊢ ( 𝜑 → 𝐽 ∈ ( Subcat ‘ 𝐶 ) )
subcss1.2 ⊢ ( 𝜑 → 𝐽 Fn ( 𝑆 × 𝑆 ) )
subcss2.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
subcss2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑆 )
subcss2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑆 )
Assertion subcss2 ( 𝜑 → ( 𝑋 𝐽 𝑌 ) ⊆ ( 𝑋 𝐻 𝑌 ) )

Proof

Step Hyp Ref Expression
1 subcss1.1 ⊢ ( 𝜑 → 𝐽 ∈ ( Subcat ‘ 𝐶 ) )
2 subcss1.2 ⊢ ( 𝜑 → 𝐽 Fn ( 𝑆 × 𝑆 ) )
3 subcss2.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
4 subcss2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑆 )
5 subcss2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑆 )
6 eqid ⊢ ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐶 )
7 1 6 subcssc ⊢ ( 𝜑 → 𝐽 ⊆cat ( Homf ‘ 𝐶 ) )
8 2 7 4 5 ssc2 ⊢ ( 𝜑 → ( 𝑋 𝐽 𝑌 ) ⊆ ( 𝑋 ( Homf ‘ 𝐶 ) 𝑌 ) )
9 eqid ⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
10 1 2 9 subcss1 ⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝐶 ) )
11 10 4 sseldd ⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐶 ) )
12 10 5 sseldd ⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐶 ) )
13 6 9 3 11 12 homfval ⊢ ( 𝜑 → ( 𝑋 ( Homf ‘ 𝐶 ) 𝑌 ) = ( 𝑋 𝐻 𝑌 ) )
14 8 13 sseqtrd ⊢ ( 𝜑 → ( 𝑋 𝐽 𝑌 ) ⊆ ( 𝑋 𝐻 𝑌 ) )