Metamath Proof Explorer

Theorem setchomfval

Description: Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017)

		Ref	Expression
	Hypotheses	setcbas.c	⊢ 𝐶 = ( SetCat ‘ 𝑈 )
		setcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
		setchomfval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	Assertion	setchomfval	⊢ ( 𝜑 → 𝐻 = ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		setcbas.c	⊢ 𝐶 = ( SetCat ‘ 𝑈 )
2		setcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
3		setchomfval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
4		eqidd	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) = ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) )
5		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 ) ) ) )
6	1 2 4 5	setcval	⊢ ( 𝜑 → 𝐶 = { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } )
7		catstr	⊢ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) ⟩ , ⟨ ( 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 ⟩
8		homid	⊢ Hom = Slot ( Hom ‘ ndx )
9		snsstp2	⊢ { ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ }
10		mpoexga	⊢ ( ( 𝑈 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉 ) → ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) ∈ V )
11	2 2 10	syl2anc	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) ∈ V )
12	6 7 8 9 11 3	strfv3	⊢ ( 𝜑 → 𝐻 = ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) )