This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem idfu2

Description: Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 28-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 )
idfu2.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
Assertion idfu2 ( 𝜑 → ( ( 𝑋 ( 2nd ‘ 𝐼 ) 𝑌 ) ‘ 𝐹 ) = 𝐹 )

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 idfu2.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
8 1 2 3 4 5 6 idfu2nd ⊢ ( 𝜑 → ( 𝑋 ( 2nd ‘ 𝐼 ) 𝑌 ) = ( I ↾ ( 𝑋 𝐻 𝑌 ) ) )
9 8 fveq1d ⊢ ( 𝜑 → ( ( 𝑋 ( 2nd ‘ 𝐼 ) 𝑌 ) ‘ 𝐹 ) = ( ( I ↾ ( 𝑋 𝐻 𝑌 ) ) ‘ 𝐹 ) )
10 fvresi ⊢ ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) → ( ( I ↾ ( 𝑋 𝐻 𝑌 ) ) ‘ 𝐹 ) = 𝐹 )
11 7 10 syl ⊢ ( 𝜑 → ( ( I ↾ ( 𝑋 𝐻 𝑌 ) ) ‘ 𝐹 ) = 𝐹 )
12 9 11 eqtrd ⊢ ( 𝜑 → ( ( 𝑋 ( 2nd ‘ 𝐼 ) 𝑌 ) ‘ 𝐹 ) = 𝐹 )