Metamath Proof Explorer

Theorem fuco2

Description: The morphism part of the functor composition bifunctor. (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 )
		fuco1.o	⊢ ( 𝜑 → ( ⟨ 𝐶 , 𝐷 ⟩ ∘_F 𝐸 ) = ⟨ 𝑂 , 𝑃 ⟩ )
		fuco1.w	⊢ ( 𝜑 → 𝑊 = ( ( 𝐷 Func 𝐸 ) × ( 𝐶 Func 𝐷 ) ) )
	Assertion	fuco2	⊢ ( 𝜑 → 𝑃 = ( 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		fuco1.o	⊢ ( 𝜑 → ( ⟨ 𝐶 , 𝐷 ⟩ ∘_F 𝐸 ) = ⟨ 𝑂 , 𝑃 ⟩ )
5		fuco1.w	⊢ ( 𝜑 → 𝑊 = ( ( 𝐷 Func 𝐸 ) × ( 𝐶 Func 𝐷 ) ) )
6	1 2 3 4 5	fucofval	⊢ ( 𝜑 → ⟨ 𝑂 , 𝑃 ⟩ = ⟨ ( ∘_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 ) ) ) ) ) ) ⟩ )
7	1 2 3 4	fucoelvv	⊢ ( 𝜑 → ⟨ 𝑂 , 𝑃 ⟩ ∈ ( V × V ) )
8		opelxp1	⊢ ( ⟨ 𝑂 , 𝑃 ⟩ ∈ ( V × V ) → 𝑂 ∈ V )
9	7 8	syl	⊢ ( 𝜑 → 𝑂 ∈ V )
10		opelxp2	⊢ ( ⟨ 𝑂 , 𝑃 ⟩ ∈ ( V × V ) → 𝑃 ∈ V )
11	7 10	syl	⊢ ( 𝜑 → 𝑃 ∈ V )
12		opthg	⊢ ( ( 𝑂 ∈ V ∧ 𝑃 ∈ V ) → ( ⟨ 𝑂 , 𝑃 ⟩ = ⟨ ( ∘_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 ) ) ) ) ) ) ⟩ ↔ ( 𝑂 = ( ∘_func ↾ 𝑊 ) ∧ 𝑃 = ( 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 ) ) ) ) ) ) ) ) )
13	9 11 12	syl2anc	⊢ ( 𝜑 → ( ⟨ 𝑂 , 𝑃 ⟩ = ⟨ ( ∘_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 ) ) ) ) ) ) ⟩ ↔ ( 𝑂 = ( ∘_func ↾ 𝑊 ) ∧ 𝑃 = ( 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 ) ) ) ) ) ) ) ) )
14	6 13	mpbid	⊢ ( 𝜑 → ( 𝑂 = ( ∘_func ↾ 𝑊 ) ∧ 𝑃 = ( 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 ) ) ) ) ) ) ) )
15	14	simprd	⊢ ( 𝜑 → 𝑃 = ( 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 ) ) ) ) ) ) )