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Metamath Proof Explorer

Theorem idfu2nd

Description: Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017)

Ref Expression
Hypotheses idfuval.i ⊢ 𝐼 = ( idfunc ‘ 𝐶 )
idfuval.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
idfuval.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
idfuval.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
idfu2nd.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
idfu2nd.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion idfu2nd ( 𝜑 → ( 𝑋 ( 2nd ‘ 𝐼 ) 𝑌 ) = ( I ↾ ( 𝑋 𝐻 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 idfuval.i ⊢ 𝐼 = ( idfunc ‘ 𝐶 )
2 idfuval.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
3 idfuval.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
4 idfuval.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
5 idfu2nd.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 idfu2nd.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 df-ov ⊢ ( 𝑋 ( 2nd ‘ 𝐼 ) 𝑌 ) = ( ( 2nd ‘ 𝐼 ) ‘ ⟨ 𝑋 , 𝑌 ⟩ )
8 1 2 3 4 idfuval ⊢ ( 𝜑 → 𝐼 = ⟨ ( I ↾ 𝐵 ) , ( 𝑧 ∈ ( 𝐵 × 𝐵 ) ↦ ( I ↾ ( 𝐻 ‘ 𝑧 ) ) ) ⟩ )
9 8 fveq2d ⊢ ( 𝜑 → ( 2nd ‘ 𝐼 ) = ( 2nd ‘ ⟨ ( I ↾ 𝐵 ) , ( 𝑧 ∈ ( 𝐵 × 𝐵 ) ↦ ( I ↾ ( 𝐻 ‘ 𝑧 ) ) ) ⟩ ) )
10 2 fvexi ⊢ 𝐵 ∈ V
11 resiexg ⊢ ( 𝐵 ∈ V → ( I ↾ 𝐵 ) ∈ V )
12 10 11 ax-mp ⊢ ( I ↾ 𝐵 ) ∈ V
13 10 10 xpex ⊢ ( 𝐵 × 𝐵 ) ∈ V
14 13 mptex ⊢ ( 𝑧 ∈ ( 𝐵 × 𝐵 ) ↦ ( I ↾ ( 𝐻 ‘ 𝑧 ) ) ) ∈ V
15 12 14 op2nd ⊢ ( 2nd ‘ ⟨ ( I ↾ 𝐵 ) , ( 𝑧 ∈ ( 𝐵 × 𝐵 ) ↦ ( I ↾ ( 𝐻 ‘ 𝑧 ) ) ) ⟩ ) = ( 𝑧 ∈ ( 𝐵 × 𝐵 ) ↦ ( I ↾ ( 𝐻 ‘ 𝑧 ) ) )
16 9 15 eqtrdi ⊢ ( 𝜑 → ( 2nd ‘ 𝐼 ) = ( 𝑧 ∈ ( 𝐵 × 𝐵 ) ↦ ( I ↾ ( 𝐻 ‘ 𝑧 ) ) ) )
17 simpr ⊢ ( ( 𝜑 ∧ 𝑧 = ⟨ 𝑋 , 𝑌 ⟩ ) → 𝑧 = ⟨ 𝑋 , 𝑌 ⟩ )
18 17 fveq2d ⊢ ( ( 𝜑 ∧ 𝑧 = ⟨ 𝑋 , 𝑌 ⟩ ) → ( 𝐻 ‘ 𝑧 ) = ( 𝐻 ‘ ⟨ 𝑋 , 𝑌 ⟩ ) )
19 df-ov ⊢ ( 𝑋 𝐻 𝑌 ) = ( 𝐻 ‘ ⟨ 𝑋 , 𝑌 ⟩ )
20 18 19 eqtr4di ⊢ ( ( 𝜑 ∧ 𝑧 = ⟨ 𝑋 , 𝑌 ⟩ ) → ( 𝐻 ‘ 𝑧 ) = ( 𝑋 𝐻 𝑌 ) )
21 20 reseq2d ⊢ ( ( 𝜑 ∧ 𝑧 = ⟨ 𝑋 , 𝑌 ⟩ ) → ( I ↾ ( 𝐻 ‘ 𝑧 ) ) = ( I ↾ ( 𝑋 𝐻 𝑌 ) ) )
22 5 6 opelxpd ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐵 × 𝐵 ) )
23 ovex ⊢ ( 𝑋 𝐻 𝑌 ) ∈ V
24 resiexg ⊢ ( ( 𝑋 𝐻 𝑌 ) ∈ V → ( I ↾ ( 𝑋 𝐻 𝑌 ) ) ∈ V )
25 23 24 mp1i ⊢ ( 𝜑 → ( I ↾ ( 𝑋 𝐻 𝑌 ) ) ∈ V )
26 16 21 22 25 fvmptd ⊢ ( 𝜑 → ( ( 2nd ‘ 𝐼 ) ‘ ⟨ 𝑋 , 𝑌 ⟩ ) = ( I ↾ ( 𝑋 𝐻 𝑌 ) ) )
27 7 26 eqtrid ⊢ ( 𝜑 → ( 𝑋 ( 2nd ‘ 𝐼 ) 𝑌 ) = ( I ↾ ( 𝑋 𝐻 𝑌 ) ) )