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Metamath Proof Explorer

Theorem idfusubc

Description: The identity functor for a subcategory is an "inclusion functor" from the subcategory into its supercategory. (Contributed by AV, 29-Mar-2020)

Ref Expression
Hypotheses idfusubc.s ⊢ 𝑆 = ( 𝐶 ↾cat 𝐽 )
idfusubc.i ⊢ 𝐼 = ( idfunc ‘ 𝑆 )
idfusubc.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
Assertion idfusubc ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → 𝐼 = ⟨ ( I ↾ 𝐵 ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 𝐽 𝑦 ) ) ) ⟩ )

Proof

Step Hyp Ref Expression
1 idfusubc.s ⊢ 𝑆 = ( 𝐶 ↾cat 𝐽 )
2 idfusubc.i ⊢ 𝐼 = ( idfunc ‘ 𝑆 )
3 idfusubc.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
4 1 2 3 idfusubc0 ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → 𝐼 = ⟨ ( I ↾ 𝐵 ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 ( Hom ‘ 𝑆 ) 𝑦 ) ) ) ⟩ )
5 eqid ⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
6 subcrcl ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → 𝐶 ∈ Cat )
7 id ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → 𝐽 ∈ ( Subcat ‘ 𝐶 ) )
8 eqidd ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → dom dom 𝐽 = dom dom 𝐽 )
9 7 8 subcfn ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → 𝐽 Fn ( dom dom 𝐽 × dom dom 𝐽 ) )
10 7 9 5 subcss1 ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → dom dom 𝐽 ⊆ ( Base ‘ 𝐶 ) )
11 1 5 6 9 10 reschom ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → 𝐽 = ( Hom ‘ 𝑆 ) )
12 11 eqcomd ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → ( Hom ‘ 𝑆 ) = 𝐽 )
13 12 oveqd ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → ( 𝑥 ( Hom ‘ 𝑆 ) 𝑦 ) = ( 𝑥 𝐽 𝑦 ) )
14 13 reseq2d ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → ( I ↾ ( 𝑥 ( Hom ‘ 𝑆 ) 𝑦 ) ) = ( I ↾ ( 𝑥 𝐽 𝑦 ) ) )
15 14 mpoeq3dv ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 ( Hom ‘ 𝑆 ) 𝑦 ) ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 𝐽 𝑦 ) ) ) )
16 15 opeq2d ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → ⟨ ( I ↾ 𝐵 ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 ( Hom ‘ 𝑆 ) 𝑦 ) ) ) ⟩ = ⟨ ( I ↾ 𝐵 ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 𝐽 𝑦 ) ) ) ⟩ )
17 4 16 eqtrd ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → 𝐼 = ⟨ ( I ↾ 𝐵 ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 𝐽 𝑦 ) ) ) ⟩ )