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Metamath Proof Explorer

Theorem idfusubc0

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 idfusubc0 ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → 𝐼 = ⟨ ( I ↾ 𝐵 ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 ( Hom ‘ 𝑆 ) 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 id ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → 𝐽 ∈ ( Subcat ‘ 𝐶 ) )
5 1 4 subccat ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → 𝑆 ∈ Cat )
6 eqid ⊢ ( Hom ‘ 𝑆 ) = ( Hom ‘ 𝑆 )
7 2 3 5 6 idfuval ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → 𝐼 = ⟨ ( I ↾ 𝐵 ) , ( 𝑧 ∈ ( 𝐵 × 𝐵 ) ↦ ( I ↾ ( ( Hom ‘ 𝑆 ) ‘ 𝑧 ) ) ) ⟩ )
8 fveq2 ⊢ ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ → ( ( Hom ‘ 𝑆 ) ‘ 𝑧 ) = ( ( Hom ‘ 𝑆 ) ‘ ⟨ 𝑥 , 𝑦 ⟩ ) )
9 df-ov ⊢ ( 𝑥 ( Hom ‘ 𝑆 ) 𝑦 ) = ( ( Hom ‘ 𝑆 ) ‘ ⟨ 𝑥 , 𝑦 ⟩ )
10 8 9 eqtr4di ⊢ ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ → ( ( Hom ‘ 𝑆 ) ‘ 𝑧 ) = ( 𝑥 ( Hom ‘ 𝑆 ) 𝑦 ) )
11 10 reseq2d ⊢ ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ → ( I ↾ ( ( Hom ‘ 𝑆 ) ‘ 𝑧 ) ) = ( I ↾ ( 𝑥 ( Hom ‘ 𝑆 ) 𝑦 ) ) )
12 11 mpompt ⊢ ( 𝑧 ∈ ( 𝐵 × 𝐵 ) ↦ ( I ↾ ( ( Hom ‘ 𝑆 ) ‘ 𝑧 ) ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 ( Hom ‘ 𝑆 ) 𝑦 ) ) )
13 12 a1i ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → ( 𝑧 ∈ ( 𝐵 × 𝐵 ) ↦ ( I ↾ ( ( Hom ‘ 𝑆 ) ‘ 𝑧 ) ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 ( Hom ‘ 𝑆 ) 𝑦 ) ) ) )
14 13 opeq2d ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → ⟨ ( I ↾ 𝐵 ) , ( 𝑧 ∈ ( 𝐵 × 𝐵 ) ↦ ( I ↾ ( ( Hom ‘ 𝑆 ) ‘ 𝑧 ) ) ) ⟩ = ⟨ ( I ↾ 𝐵 ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 ( Hom ‘ 𝑆 ) 𝑦 ) ) ) ⟩ )
15 7 14 eqtrd ⊢ ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) → 𝐼 = ⟨ ( I ↾ 𝐵 ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 ( Hom ‘ 𝑆 ) 𝑦 ) ) ) ⟩ )