Theorem tposcurf2cl

Description: The transposed curry functor at a morphism is a 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.n	⊢ 𝑁 = ( 𝐷 Nat 𝐸 )
Assertion	tposcurf2cl	⊢ ( 𝜑 → 𝐿 ∈ ( ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) 𝑁 ( ( 1st ‘ 𝐺 ) ‘ 𝑌 ) ) )

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.n	⊢ 𝑁 = ( 𝐷 Nat 𝐸 )
14		eqid	⊢ ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ) = ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) )
15		eqidd	⊢ ( 𝜑 → ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) = ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) )
16	3 4 5 15	cofuswapfcl	⊢ ( 𝜑 → ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) )
17		eqid	⊢ ( ( 𝑋 ( 2nd ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ) ) 𝑌 ) ‘ 𝐾 ) = ( ( 𝑋 ( 2nd ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ) ) 𝑌 ) ‘ 𝐾 )
18	14 2 3 4 16 6 7 8 9 10 11 17 13	curf2cl	⊢ ( 𝜑 → ( ( 𝑋 ( 2nd ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ) ) 𝑌 ) ‘ 𝐾 ) ∈ ( ( ( 1st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ) ) ‘ 𝑋 ) 𝑁 ( ( 1st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ) ) ‘ 𝑌 ) ) )
19	1	fveq2d	⊢ ( 𝜑 → ( 2nd ‘ 𝐺 ) = ( 2nd ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ) ) )
20	19	oveqd	⊢ ( 𝜑 → ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑌 ) = ( 𝑋 ( 2nd ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ) ) 𝑌 ) )
21	20	fveq1d	⊢ ( 𝜑 → ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑌 ) ‘ 𝐾 ) = ( ( 𝑋 ( 2nd ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ) ) 𝑌 ) ‘ 𝐾 ) )
22	12 21	eqtrd	⊢ ( 𝜑 → 𝐿 = ( ( 𝑋 ( 2nd ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ) ) 𝑌 ) ‘ 𝐾 ) )
23	1	fveq2d	⊢ ( 𝜑 → ( 1st ‘ 𝐺 ) = ( 1st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ) ) )
24	23	fveq1d	⊢ ( 𝜑 → ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) = ( ( 1st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ) ) ‘ 𝑋 ) )
25	23	fveq1d	⊢ ( 𝜑 → ( ( 1st ‘ 𝐺 ) ‘ 𝑌 ) = ( ( 1st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ) ) ‘ 𝑌 ) )
26	24 25	oveq12d	⊢ ( 𝜑 → ( ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) 𝑁 ( ( 1st ‘ 𝐺 ) ‘ 𝑌 ) ) = ( ( ( 1st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ) ) ‘ 𝑋 ) 𝑁 ( ( 1st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐹 ∘func ( 𝐶 swapF 𝐷 ) ) ) ) ‘ 𝑌 ) ) )
27	18 22 26	3eltr4d	⊢ ( 𝜑 → 𝐿 ∈ ( ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) 𝑁 ( ( 1st ‘ 𝐺 ) ‘ 𝑌 ) ) )