Metamath Proof Explorer


Theorem prf1

Description: Value of the pairing functor on objects. (Contributed by Mario Carneiro, 12-Jan-2017)

Ref Expression
Hypotheses prfval.k 𝑃 = ( 𝐹 ⟨,⟩F 𝐺 )
prfval.b 𝐵 = ( Base ‘ 𝐶 )
prfval.h 𝐻 = ( Hom ‘ 𝐶 )
prfval.c ( 𝜑𝐹 ∈ ( 𝐶 Func 𝐷 ) )
prfval.d ( 𝜑𝐺 ∈ ( 𝐶 Func 𝐸 ) )
prf1.x ( 𝜑𝑋𝐵 )
Assertion prf1 ( 𝜑 → ( ( 1st𝑃 ) ‘ 𝑋 ) = ⟨ ( ( 1st𝐹 ) ‘ 𝑋 ) , ( ( 1st𝐺 ) ‘ 𝑋 ) ⟩ )

Proof

Step Hyp Ref Expression
1 prfval.k 𝑃 = ( 𝐹 ⟨,⟩F 𝐺 )
2 prfval.b 𝐵 = ( Base ‘ 𝐶 )
3 prfval.h 𝐻 = ( Hom ‘ 𝐶 )
4 prfval.c ( 𝜑𝐹 ∈ ( 𝐶 Func 𝐷 ) )
5 prfval.d ( 𝜑𝐺 ∈ ( 𝐶 Func 𝐸 ) )
6 prf1.x ( 𝜑𝑋𝐵 )
7 1 2 3 4 5 prfval ( 𝜑𝑃 = ⟨ ( 𝑥𝐵 ↦ ⟨ ( ( 1st𝐹 ) ‘ 𝑥 ) , ( ( 1st𝐺 ) ‘ 𝑥 ) ⟩ ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( ∈ ( 𝑥 𝐻 𝑦 ) ↦ ⟨ ( ( 𝑥 ( 2nd𝐹 ) 𝑦 ) ‘ ) , ( ( 𝑥 ( 2nd𝐺 ) 𝑦 ) ‘ ) ⟩ ) ) ⟩ )
8 2 fvexi 𝐵 ∈ V
9 8 mptex ( 𝑥𝐵 ↦ ⟨ ( ( 1st𝐹 ) ‘ 𝑥 ) , ( ( 1st𝐺 ) ‘ 𝑥 ) ⟩ ) ∈ V
10 8 8 mpoex ( 𝑥𝐵 , 𝑦𝐵 ↦ ( ∈ ( 𝑥 𝐻 𝑦 ) ↦ ⟨ ( ( 𝑥 ( 2nd𝐹 ) 𝑦 ) ‘ ) , ( ( 𝑥 ( 2nd𝐺 ) 𝑦 ) ‘ ) ⟩ ) ) ∈ V
11 9 10 op1std ( 𝑃 = ⟨ ( 𝑥𝐵 ↦ ⟨ ( ( 1st𝐹 ) ‘ 𝑥 ) , ( ( 1st𝐺 ) ‘ 𝑥 ) ⟩ ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( ∈ ( 𝑥 𝐻 𝑦 ) ↦ ⟨ ( ( 𝑥 ( 2nd𝐹 ) 𝑦 ) ‘ ) , ( ( 𝑥 ( 2nd𝐺 ) 𝑦 ) ‘ ) ⟩ ) ) ⟩ → ( 1st𝑃 ) = ( 𝑥𝐵 ↦ ⟨ ( ( 1st𝐹 ) ‘ 𝑥 ) , ( ( 1st𝐺 ) ‘ 𝑥 ) ⟩ ) )
12 7 11 syl ( 𝜑 → ( 1st𝑃 ) = ( 𝑥𝐵 ↦ ⟨ ( ( 1st𝐹 ) ‘ 𝑥 ) , ( ( 1st𝐺 ) ‘ 𝑥 ) ⟩ ) )
13 simpr ( ( 𝜑𝑥 = 𝑋 ) → 𝑥 = 𝑋 )
14 13 fveq2d ( ( 𝜑𝑥 = 𝑋 ) → ( ( 1st𝐹 ) ‘ 𝑥 ) = ( ( 1st𝐹 ) ‘ 𝑋 ) )
15 13 fveq2d ( ( 𝜑𝑥 = 𝑋 ) → ( ( 1st𝐺 ) ‘ 𝑥 ) = ( ( 1st𝐺 ) ‘ 𝑋 ) )
16 14 15 opeq12d ( ( 𝜑𝑥 = 𝑋 ) → ⟨ ( ( 1st𝐹 ) ‘ 𝑥 ) , ( ( 1st𝐺 ) ‘ 𝑥 ) ⟩ = ⟨ ( ( 1st𝐹 ) ‘ 𝑋 ) , ( ( 1st𝐺 ) ‘ 𝑋 ) ⟩ )
17 opex ⟨ ( ( 1st𝐹 ) ‘ 𝑋 ) , ( ( 1st𝐺 ) ‘ 𝑋 ) ⟩ ∈ V
18 17 a1i ( 𝜑 → ⟨ ( ( 1st𝐹 ) ‘ 𝑋 ) , ( ( 1st𝐺 ) ‘ 𝑋 ) ⟩ ∈ V )
19 12 16 6 18 fvmptd ( 𝜑 → ( ( 1st𝑃 ) ‘ 𝑋 ) = ⟨ ( ( 1st𝐹 ) ‘ 𝑋 ) , ( ( 1st𝐺 ) ‘ 𝑋 ) ⟩ )