Theorem rngcbasALTV

Description: Set of objects 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 )
Assertion	rngcbasALTV	⊢ ( 𝜑 → 𝐵 = ( 𝑈 ∩ Rng ) )

Proof

Step	Hyp	Ref	Expression
1		rngcbasALTV.c	⊢ 𝐶 = ( RngCatALTV ‘ 𝑈 )
2		rngcbasALTV.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		rngcbasALTV.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
4		eqidd	⊢ ( 𝜑 → ( 𝑈 ∩ Rng ) = ( 𝑈 ∩ Rng ) )
5		eqidd	⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝑈 ∩ Rng ) , 𝑦 ∈ ( 𝑈 ∩ Rng ) ↦ ( 𝑥 RngHom 𝑦 ) ) = ( 𝑥 ∈ ( 𝑈 ∩ Rng ) , 𝑦 ∈ ( 𝑈 ∩ Rng ) ↦ ( 𝑥 RngHom 𝑦 ) ) )
6		eqidd	⊢ ( 𝜑 → ( 𝑣 ∈ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) , 𝑧 ∈ ( 𝑈 ∩ Rng ) ↦ ( 𝑓 ∈ ( ( 2nd ‘ 𝑣 ) RngHom 𝑧 ) , 𝑔 ∈ ( ( 1st ‘ 𝑣 ) RngHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑓 ∘ 𝑔 ) ) ) = ( 𝑣 ∈ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) , 𝑧 ∈ ( 𝑈 ∩ Rng ) ↦ ( 𝑓 ∈ ( ( 2nd ‘ 𝑣 ) RngHom 𝑧 ) , 𝑔 ∈ ( ( 1st ‘ 𝑣 ) RngHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑓 ∘ 𝑔 ) ) ) )
7	1 3 4 5 6	rngcvalALTV	⊢ ( 𝜑 → 𝐶 = { ⟨ ( Base ‘ ndx ) , ( 𝑈 ∩ Rng ) ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ ( 𝑈 ∩ Rng ) , 𝑦 ∈ ( 𝑈 ∩ Rng ) ↦ ( 𝑥 RngHom 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) , 𝑧 ∈ ( 𝑈 ∩ Rng ) ↦ ( 𝑓 ∈ ( ( 2nd ‘ 𝑣 ) RngHom 𝑧 ) , 𝑔 ∈ ( ( 1st ‘ 𝑣 ) RngHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑓 ∘ 𝑔 ) ) ) ⟩ } )
8		catstr	⊢ { ⟨ ( Base ‘ ndx ) , ( 𝑈 ∩ Rng ) ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ ( 𝑈 ∩ Rng ) , 𝑦 ∈ ( 𝑈 ∩ Rng ) ↦ ( 𝑥 RngHom 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) , 𝑧 ∈ ( 𝑈 ∩ Rng ) ↦ ( 𝑓 ∈ ( ( 2nd ‘ 𝑣 ) RngHom 𝑧 ) , 𝑔 ∈ ( ( 1st ‘ 𝑣 ) RngHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑓 ∘ 𝑔 ) ) ) ⟩ } Struct ⟨ 1 , ; 1 5 ⟩
9		baseid	⊢ Base = Slot ( Base ‘ ndx )
10		snsstp1	⊢ { ⟨ ( Base ‘ ndx ) , ( 𝑈 ∩ Rng ) ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , ( 𝑈 ∩ Rng ) ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ ( 𝑈 ∩ Rng ) , 𝑦 ∈ ( 𝑈 ∩ Rng ) ↦ ( 𝑥 RngHom 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) , 𝑧 ∈ ( 𝑈 ∩ Rng ) ↦ ( 𝑓 ∈ ( ( 2nd ‘ 𝑣 ) RngHom 𝑧 ) , 𝑔 ∈ ( ( 1st ‘ 𝑣 ) RngHom ( 2nd ‘ 𝑣 ) ) ↦ ( 𝑓 ∘ 𝑔 ) ) ) ⟩ }
11		inex1g	⊢ ( 𝑈 ∈ 𝑉 → ( 𝑈 ∩ Rng ) ∈ V )
12	3 11	syl	⊢ ( 𝜑 → ( 𝑈 ∩ Rng ) ∈ V )
13	7 8 9 10 12 2	strfv3	⊢ ( 𝜑 → 𝐵 = ( 𝑈 ∩ Rng ) )