Theorem tposcurf2val

Description: Value of a component of the transposed curry functor natural transformation. (Contributed by Zhi Wang, 10-Oct-2025)

	Ref	Expression
Hypotheses	tposcurf2.g	⊢ ( 𝜑 → 𝐺 = ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ) )
	tposcurf2.a	⊢ 𝐴 = ( Base ‘ 𝐶 )
	tposcurf2.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	tposcurf2.d	⊢ ( 𝜑 → 𝐷 ∈ Cat )
	tposcurf2.f	⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐷 ×c 𝐶 ) Func 𝐸 ) )
	tposcurf2.b	⊢ 𝐵 = ( Base ‘ 𝐷 )
	tposcurf2.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	tposcurf2.i	⊢ 𝐼 = ( Id ‘ 𝐷 )
	tposcurf2.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
	tposcurf2.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐴 )
	tposcurf2.k	⊢ ( 𝜑 → 𝐾 ∈ ( 𝑋 𝐻 𝑌 ) )
	tposcurf2.l	⊢ ( 𝜑 → 𝐿 = ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑌 ) ‘ 𝐾 ) )
	tposcurf2.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
Assertion	tposcurf2val	⊢ ( 𝜑 → ( 𝐿 ‘ 𝑍 ) = ( ( 𝐼 ‘ 𝑍 ) ( ⟨ 𝑍 , 𝑋 ⟩ ( 2nd ‘ 𝐹 ) ⟨ 𝑍 , 𝑌 ⟩ ) 𝐾 ) )

Proof

Step	Hyp	Ref	Expression
1		tposcurf2.g	⊢ ( 𝜑 → 𝐺 = ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ) )
2		tposcurf2.a	⊢ 𝐴 = ( Base ‘ 𝐶 )
3		tposcurf2.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
4		tposcurf2.d	⊢ ( 𝜑 → 𝐷 ∈ Cat )
5		tposcurf2.f	⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐷 ×c 𝐶 ) Func 𝐸 ) )
6		tposcurf2.b	⊢ 𝐵 = ( Base ‘ 𝐷 )
7		tposcurf2.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
8		tposcurf2.i	⊢ 𝐼 = ( Id ‘ 𝐷 )
9		tposcurf2.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
10		tposcurf2.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐴 )
11		tposcurf2.k	⊢ ( 𝜑 → 𝐾 ∈ ( 𝑋 𝐻 𝑌 ) )
12		tposcurf2.l	⊢ ( 𝜑 → 𝐿 = ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑌 ) ‘ 𝐾 ) )
13		tposcurf2.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
14	1 2 3 4 5 6 7 8 9 10 11 12	tposcurf2	⊢ ( 𝜑 → 𝐿 = ( 𝑧 ∈ 𝐵 ↦ ( ( 𝐼 ‘ 𝑧 ) ( ⟨ 𝑧 , 𝑋 ⟩ ( 2nd ‘ 𝐹 ) ⟨ 𝑧 , 𝑌 ⟩ ) 𝐾 ) ) )
15		simpr	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑍 ) → 𝑧 = 𝑍 )
16	15	opeq1d	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑍 ) → ⟨ 𝑧 , 𝑋 ⟩ = ⟨ 𝑍 , 𝑋 ⟩ )
17	15	opeq1d	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑍 ) → ⟨ 𝑧 , 𝑌 ⟩ = ⟨ 𝑍 , 𝑌 ⟩ )
18	16 17	oveq12d	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑍 ) → ( ⟨ 𝑧 , 𝑋 ⟩ ( 2nd ‘ 𝐹 ) ⟨ 𝑧 , 𝑌 ⟩ ) = ( ⟨ 𝑍 , 𝑋 ⟩ ( 2nd ‘ 𝐹 ) ⟨ 𝑍 , 𝑌 ⟩ ) )
19	15	fveq2d	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑍 ) → ( 𝐼 ‘ 𝑧 ) = ( 𝐼 ‘ 𝑍 ) )
20		eqidd	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑍 ) → 𝐾 = 𝐾 )
21	18 19 20	oveq123d	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑍 ) → ( ( 𝐼 ‘ 𝑧 ) ( ⟨ 𝑧 , 𝑋 ⟩ ( 2nd ‘ 𝐹 ) ⟨ 𝑧 , 𝑌 ⟩ ) 𝐾 ) = ( ( 𝐼 ‘ 𝑍 ) ( ⟨ 𝑍 , 𝑋 ⟩ ( 2nd ‘ 𝐹 ) ⟨ 𝑍 , 𝑌 ⟩ ) 𝐾 ) )
22		ovexd	⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑍 ) ( ⟨ 𝑍 , 𝑋 ⟩ ( 2nd ‘ 𝐹 ) ⟨ 𝑍 , 𝑌 ⟩ ) 𝐾 ) ∈ V )
23	14 21 13 22	fvmptd	⊢ ( 𝜑 → ( 𝐿 ‘ 𝑍 ) = ( ( 𝐼 ‘ 𝑍 ) ( ⟨ 𝑍 , 𝑋 ⟩ ( 2nd ‘ 𝐹 ) ⟨ 𝑍 , 𝑌 ⟩ ) 𝐾 ) )