tposcurf11

Description: Value of the double evaluated transposed curry functor. (Contributed by Zhi Wang, 9-Oct-2025)

	Ref	Expression
Hypotheses	tposcurf1.g	⊢ ( 𝜑 → 𝐺 = ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐹 ∘_func ( 𝐶 swap_F 𝐷 ) ) ) )
	tposcurf1.a	⊢ 𝐴 = ( Base ‘ 𝐶 )
	tposcurf1.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	tposcurf1.d	⊢ ( 𝜑 → 𝐷 ∈ Cat )
	tposcurf1.f	⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐷 ×_c 𝐶 ) Func 𝐸 ) )
	tposcurf1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
	tposcurf1.k	⊢ ( 𝜑 → 𝐾 = ( ( 1^st ‘ 𝐺 ) ‘ 𝑋 ) )
	tposcurf1.b	⊢ 𝐵 = ( Base ‘ 𝐷 )
	tposcurf11.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	tposcurf11	⊢ ( 𝜑 → ( ( 1^st ‘ 𝐾 ) ‘ 𝑌 ) = ( 𝑌 ( 1^st ‘ 𝐹 ) 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		tposcurf1.g	⊢ ( 𝜑 → 𝐺 = ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐹 ∘_func ( 𝐶 swap_F 𝐷 ) ) ) )
2		tposcurf1.a	⊢ 𝐴 = ( Base ‘ 𝐶 )
3		tposcurf1.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
4		tposcurf1.d	⊢ ( 𝜑 → 𝐷 ∈ Cat )
5		tposcurf1.f	⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐷 ×_c 𝐶 ) Func 𝐸 ) )
6		tposcurf1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
7		tposcurf1.k	⊢ ( 𝜑 → 𝐾 = ( ( 1^st ‘ 𝐺 ) ‘ 𝑋 ) )
8		tposcurf1.b	⊢ 𝐵 = ( Base ‘ 𝐷 )
9		tposcurf11.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
10	1	fveq2d	⊢ ( 𝜑 → ( 1^st ‘ 𝐺 ) = ( 1^st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐹 ∘_func ( 𝐶 swap_F 𝐷 ) ) ) ) )
11	10	fveq1d	⊢ ( 𝜑 → ( ( 1^st ‘ 𝐺 ) ‘ 𝑋 ) = ( ( 1^st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐹 ∘_func ( 𝐶 swap_F 𝐷 ) ) ) ) ‘ 𝑋 ) )
12	7 11	eqtrd	⊢ ( 𝜑 → 𝐾 = ( ( 1^st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐹 ∘_func ( 𝐶 swap_F 𝐷 ) ) ) ) ‘ 𝑋 ) )
13	12	fveq2d	⊢ ( 𝜑 → ( 1^st ‘ 𝐾 ) = ( 1^st ‘ ( ( 1^st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐹 ∘_func ( 𝐶 swap_F 𝐷 ) ) ) ) ‘ 𝑋 ) ) )
14	13	fveq1d	⊢ ( 𝜑 → ( ( 1^st ‘ 𝐾 ) ‘ 𝑌 ) = ( ( 1^st ‘ ( ( 1^st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐹 ∘_func ( 𝐶 swap_F 𝐷 ) ) ) ) ‘ 𝑋 ) ) ‘ 𝑌 ) )
15		eqid	⊢ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐹 ∘_func ( 𝐶 swap_F 𝐷 ) ) ) = ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐹 ∘_func ( 𝐶 swap_F 𝐷 ) ) )
16		eqidd	⊢ ( 𝜑 → ( 𝐹 ∘_func ( 𝐶 swap_F 𝐷 ) ) = ( 𝐹 ∘_func ( 𝐶 swap_F 𝐷 ) ) )
17	3 4 5 16	cofuswapfcl	⊢ ( 𝜑 → ( 𝐹 ∘_func ( 𝐶 swap_F 𝐷 ) ) ∈ ( ( 𝐶 ×_c 𝐷 ) Func 𝐸 ) )
18		eqid	⊢ ( ( 1^st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐹 ∘_func ( 𝐶 swap_F 𝐷 ) ) ) ) ‘ 𝑋 ) = ( ( 1^st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐹 ∘_func ( 𝐶 swap_F 𝐷 ) ) ) ) ‘ 𝑋 )
19	15 2 3 4 17 8 6 18 9	curf11	⊢ ( 𝜑 → ( ( 1^st ‘ ( ( 1^st ‘ ( ⟨ 𝐶 , 𝐷 ⟩ curry_F ( 𝐹 ∘_func ( 𝐶 swap_F 𝐷 ) ) ) ) ‘ 𝑋 ) ) ‘ 𝑌 ) = ( 𝑋 ( 1^st ‘ ( 𝐹 ∘_func ( 𝐶 swap_F 𝐷 ) ) ) 𝑌 ) )
20	3 4 5 16 2 8 6 9	cofuswapf1	⊢ ( 𝜑 → ( 𝑋 ( 1^st ‘ ( 𝐹 ∘_func ( 𝐶 swap_F 𝐷 ) ) ) 𝑌 ) = ( 𝑌 ( 1^st ‘ 𝐹 ) 𝑋 ) )
21	14 19 20	3eqtrd	⊢ ( 𝜑 → ( ( 1^st ‘ 𝐾 ) ‘ 𝑌 ) = ( 𝑌 ( 1^st ‘ 𝐹 ) 𝑋 ) )

Metamath Proof Explorer

Theorem tposcurf11

Proof