Metamath Proof Explorer

Theorem fucofval

Description: Value of the function giving the functor composition bifunctor. Hypotheses fucofval.c and fucofval.d are not redundant ( fucofvalne ). (Contributed by Zhi Wang, 29-Sep-2025)

		Ref	Expression
	Hypotheses	fucofval.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑇 )
		fucofval.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑈 )
		fucofval.e	⊢ ( 𝜑 → 𝐸 ∈ 𝑉 )
		fucofval.o	⊢ ( 𝜑 → ( ⟨ 𝐶 , 𝐷 ⟩ ∘_F 𝐸 ) = ⚬ )
		fucofval.w	⊢ ( 𝜑 → 𝑊 = ( ( 𝐷 Func 𝐸 ) × ( 𝐶 Func 𝐷 ) ) )
	Assertion	fucofval	⊢ ( 𝜑 → ⚬ = ⟨ ( ∘_func ↾ 𝑊 ) , ( 𝑢 ∈ 𝑊 , 𝑣 ∈ 𝑊 ↦ ⦋ ( 1^st ‘ ( 2^nd ‘ 𝑢 ) ) / 𝑓 ⦌ ⦋ ( 1^st ‘ ( 1^st ‘ 𝑢 ) ) / 𝑘 ⦌ ⦋ ( 2^nd ‘ ( 1^st ‘ 𝑢 ) ) / 𝑙 ⦌ ⦋ ( 1^st ‘ ( 2^nd ‘ 𝑣 ) ) / 𝑚 ⦌ ⦋ ( 1^st ‘ ( 1^st ‘ 𝑣 ) ) / 𝑟 ⦌ ( 𝑏 ∈ ( ( 1^st ‘ 𝑢 ) ( 𝐷 Nat 𝐸 ) ( 1^st ‘ 𝑣 ) ) , 𝑎 ∈ ( ( 2^nd ‘ 𝑢 ) ( 𝐶 Nat 𝐷 ) ( 2^nd ‘ 𝑣 ) ) ↦ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ ( ( 𝑏 ‘ ( 𝑚 ‘ 𝑥 ) ) ( ⟨ ( 𝑘 ‘ ( 𝑓 ‘ 𝑥 ) ) , ( 𝑘 ‘ ( 𝑚 ‘ 𝑥 ) ) ⟩ ( comp ‘ 𝐸 ) ( 𝑟 ‘ ( 𝑚 ‘ 𝑥 ) ) ) ( ( ( 𝑓 ‘ 𝑥 ) 𝑙 ( 𝑚 ‘ 𝑥 ) ) ‘ ( 𝑎 ‘ 𝑥 ) ) ) ) ) ) ⟩ )

Proof

Step	Hyp	Ref	Expression
1		fucofval.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑇 )
2		fucofval.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑈 )
3		fucofval.e	⊢ ( 𝜑 → 𝐸 ∈ 𝑉 )
4		fucofval.o	⊢ ( 𝜑 → ( ⟨ 𝐶 , 𝐷 ⟩ ∘_F 𝐸 ) = ⚬ )
5		fucofval.w	⊢ ( 𝜑 → 𝑊 = ( ( 𝐷 Func 𝐸 ) × ( 𝐶 Func 𝐷 ) ) )
6		opex	⊢ ⟨ 𝐶 , 𝐷 ⟩ ∈ V
7	6	a1i	⊢ ( 𝜑 → ⟨ 𝐶 , 𝐷 ⟩ ∈ V )
8		op1stg	⊢ ( ( 𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈 ) → ( 1^st ‘ ⟨ 𝐶 , 𝐷 ⟩ ) = 𝐶 )
9	1 2 8	syl2anc	⊢ ( 𝜑 → ( 1^st ‘ ⟨ 𝐶 , 𝐷 ⟩ ) = 𝐶 )
10		op2ndg	⊢ ( ( 𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈 ) → ( 2^nd ‘ ⟨ 𝐶 , 𝐷 ⟩ ) = 𝐷 )
11	1 2 10	syl2anc	⊢ ( 𝜑 → ( 2^nd ‘ ⟨ 𝐶 , 𝐷 ⟩ ) = 𝐷 )
12	7 9 11 3 4 5	fucofvalg	⊢ ( 𝜑 → ⚬ = ⟨ ( ∘_func ↾ 𝑊 ) , ( 𝑢 ∈ 𝑊 , 𝑣 ∈ 𝑊 ↦ ⦋ ( 1^st ‘ ( 2^nd ‘ 𝑢 ) ) / 𝑓 ⦌ ⦋ ( 1^st ‘ ( 1^st ‘ 𝑢 ) ) / 𝑘 ⦌ ⦋ ( 2^nd ‘ ( 1^st ‘ 𝑢 ) ) / 𝑙 ⦌ ⦋ ( 1^st ‘ ( 2^nd ‘ 𝑣 ) ) / 𝑚 ⦌ ⦋ ( 1^st ‘ ( 1^st ‘ 𝑣 ) ) / 𝑟 ⦌ ( 𝑏 ∈ ( ( 1^st ‘ 𝑢 ) ( 𝐷 Nat 𝐸 ) ( 1^st ‘ 𝑣 ) ) , 𝑎 ∈ ( ( 2^nd ‘ 𝑢 ) ( 𝐶 Nat 𝐷 ) ( 2^nd ‘ 𝑣 ) ) ↦ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ ( ( 𝑏 ‘ ( 𝑚 ‘ 𝑥 ) ) ( ⟨ ( 𝑘 ‘ ( 𝑓 ‘ 𝑥 ) ) , ( 𝑘 ‘ ( 𝑚 ‘ 𝑥 ) ) ⟩ ( comp ‘ 𝐸 ) ( 𝑟 ‘ ( 𝑚 ‘ 𝑥 ) ) ) ( ( ( 𝑓 ‘ 𝑥 ) 𝑙 ( 𝑚 ‘ 𝑥 ) ) ‘ ( 𝑎 ‘ 𝑥 ) ) ) ) ) ) ⟩ )