Metamath Proof Explorer

Theorem rngchomfvalALTV

Description: Set of arrows of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020)

		Ref	Expression
	Hypotheses	rngcbasALTV.c	⊢ 𝐶 = ( RngCatALTV ‘ 𝑈 )
		rngcbasALTV.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
		rngcbasALTV.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
		rngchomfvalALTV.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	Assertion	rngchomfvalALTV	⊢ ( 𝜑 → 𝐻 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 RngHom 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rngcbasALTV.c	⊢ 𝐶 = ( RngCatALTV ‘ 𝑈 )
2		rngcbasALTV.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		rngcbasALTV.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
4		rngchomfvalALTV.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
5	1 2 3	rngcbasALTV	⊢ ( 𝜑 → 𝐵 = ( 𝑈 ∩ Rng ) )
6		eqidd	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 RngHom 𝑦 ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 RngHom 𝑦 ) ) )
7		eqidd	⊢ ( 𝜑 → ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) RngHom 𝑧 ) , 𝑔 ∈ ( ( 1^st ‘ 𝑣 ) RngHom ( 2^nd ‘ 𝑣 ) ) ↦ ( 𝑓 ∘ 𝑔 ) ) ) = ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) RngHom 𝑧 ) , 𝑔 ∈ ( ( 1^st ‘ 𝑣 ) RngHom ( 2^nd ‘ 𝑣 ) ) ↦ ( 𝑓 ∘ 𝑔 ) ) ) )
8	1 3 5 6 7	rngcvalALTV	⊢ ( 𝜑 → 𝐶 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 RngHom 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) RngHom 𝑧 ) , 𝑔 ∈ ( ( 1^st ‘ 𝑣 ) RngHom ( 2^nd ‘ 𝑣 ) ) ↦ ( 𝑓 ∘ 𝑔 ) ) ) ⟩ } )
9	8	fveq2d	⊢ ( 𝜑 → ( Hom ‘ 𝐶 ) = ( Hom ‘ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 RngHom 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) RngHom 𝑧 ) , 𝑔 ∈ ( ( 1^st ‘ 𝑣 ) RngHom ( 2^nd ‘ 𝑣 ) ) ↦ ( 𝑓 ∘ 𝑔 ) ) ) ⟩ } ) )
10	4 9	eqtrid	⊢ ( 𝜑 → 𝐻 = ( Hom ‘ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 RngHom 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) RngHom 𝑧 ) , 𝑔 ∈ ( ( 1^st ‘ 𝑣 ) RngHom ( 2^nd ‘ 𝑣 ) ) ↦ ( 𝑓 ∘ 𝑔 ) ) ) ⟩ } ) )
11	2	fvexi	⊢ 𝐵 ∈ V
12	11 11	mpoex	⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 RngHom 𝑦 ) ) ∈ V
13		catstr	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 RngHom 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) RngHom 𝑧 ) , 𝑔 ∈ ( ( 1^st ‘ 𝑣 ) RngHom ( 2^nd ‘ 𝑣 ) ) ↦ ( 𝑓 ∘ 𝑔 ) ) ) ⟩ } Struct ⟨ 1 , ; 1 5 ⟩
14		homid	⊢ Hom = Slot ( Hom ‘ ndx )
15		snsstp2	⊢ { ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 RngHom 𝑦 ) ) ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 RngHom 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) RngHom 𝑧 ) , 𝑔 ∈ ( ( 1^st ‘ 𝑣 ) RngHom ( 2^nd ‘ 𝑣 ) ) ↦ ( 𝑓 ∘ 𝑔 ) ) ) ⟩ }
16	13 14 15	strfv	⊢ ( ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 RngHom 𝑦 ) ) ∈ V → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 RngHom 𝑦 ) ) = ( Hom ‘ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 RngHom 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) RngHom 𝑧 ) , 𝑔 ∈ ( ( 1^st ‘ 𝑣 ) RngHom ( 2^nd ‘ 𝑣 ) ) ↦ ( 𝑓 ∘ 𝑔 ) ) ) ⟩ } ) )
17	12 16	mp1i	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 RngHom 𝑦 ) ) = ( Hom ‘ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 RngHom 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) RngHom 𝑧 ) , 𝑔 ∈ ( ( 1^st ‘ 𝑣 ) RngHom ( 2^nd ‘ 𝑣 ) ) ↦ ( 𝑓 ∘ 𝑔 ) ) ) ⟩ } ) )
18	10 17	eqtr4d	⊢ ( 𝜑 → 𝐻 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 RngHom 𝑦 ) ) )