Metamath Proof Explorer

Theorem setccofval

Description: Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017)

		Ref	Expression
	Hypotheses	setcbas.c	⊢ 𝐶 = ( SetCat ‘ 𝑈 )
		setcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
		setcco.o	⊢ · = ( comp ‘ 𝐶 )
	Assertion	setccofval	⊢ ( 𝜑 → · = ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		setcbas.c	⊢ 𝐶 = ( SetCat ‘ 𝑈 )
2		setcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
3		setcco.o	⊢ · = ( comp ‘ 𝐶 )
4		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
5	1 2 4	setchomfval	⊢ ( 𝜑 → ( Hom ‘ 𝐶 ) = ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) )
6		eqidd	⊢ ( 𝜑 → ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) = ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) )
7	1 2 5 6	setcval	⊢ ( 𝜑 → 𝐶 = { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } )
8		catstr	⊢ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } Struct ⟨ 1 , ; 1 5 ⟩
9		ccoid	⊢ comp = Slot ( comp ‘ ndx )
10		snsstp3	⊢ { ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ }
11	2 2	xpexd	⊢ ( 𝜑 → ( 𝑈 × 𝑈 ) ∈ V )
12		mpoexga	⊢ ( ( ( 𝑈 × 𝑈 ) ∈ V ∧ 𝑈 ∈ 𝑉 ) → ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ∈ V )
13	11 2 12	syl2anc	⊢ ( 𝜑 → ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ∈ V )
14	7 8 9 10 13 3	strfv3	⊢ ( 𝜑 → · = ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) )