Theorem estrchomfval

Description: Set of morphisms ("arrows") of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020)

	Ref	Expression
Hypotheses	estrcbas.c	⊢ 𝐶 = ( ExtStrCat ‘ 𝑈 )
	estrcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
	estrchomfval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
Assertion	estrchomfval	⊢ ( 𝜑 → 𝐻 = ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) )

Proof

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