Metamath Proof Explorer


Theorem ringcbasALTV

Description: Set of objects of the category of rings (in a universe). (Contributed by AV, 13-Feb-2020) (New usage is discouraged.)

Ref Expression
Hypotheses ringcbasALTV.c 𝐶 = ( RingCatALTV ‘ 𝑈 )
ringcbasALTV.b 𝐵 = ( Base ‘ 𝐶 )
ringcbasALTV.u ( 𝜑𝑈𝑉 )
Assertion ringcbasALTV ( 𝜑𝐵 = ( 𝑈 ∩ Ring ) )

Proof

Step Hyp Ref Expression
1 ringcbasALTV.c 𝐶 = ( RingCatALTV ‘ 𝑈 )
2 ringcbasALTV.b 𝐵 = ( Base ‘ 𝐶 )
3 ringcbasALTV.u ( 𝜑𝑈𝑉 )
4 eqidd ( 𝜑 → ( 𝑈 ∩ Ring ) = ( 𝑈 ∩ Ring ) )
5 eqidd ( 𝜑 → ( 𝑥 ∈ ( 𝑈 ∩ Ring ) , 𝑦 ∈ ( 𝑈 ∩ Ring ) ↦ ( 𝑥 RingHom 𝑦 ) ) = ( 𝑥 ∈ ( 𝑈 ∩ Ring ) , 𝑦 ∈ ( 𝑈 ∩ Ring ) ↦ ( 𝑥 RingHom 𝑦 ) ) )
6 eqidd ( 𝜑 → ( 𝑣 ∈ ( ( 𝑈 ∩ Ring ) × ( 𝑈 ∩ Ring ) ) , 𝑧 ∈ ( 𝑈 ∩ Ring ) ↦ ( 𝑓 ∈ ( ( 2nd𝑣 ) RingHom 𝑧 ) , 𝑔 ∈ ( ( 1st𝑣 ) RingHom ( 2nd𝑣 ) ) ↦ ( 𝑓𝑔 ) ) ) = ( 𝑣 ∈ ( ( 𝑈 ∩ Ring ) × ( 𝑈 ∩ Ring ) ) , 𝑧 ∈ ( 𝑈 ∩ Ring ) ↦ ( 𝑓 ∈ ( ( 2nd𝑣 ) RingHom 𝑧 ) , 𝑔 ∈ ( ( 1st𝑣 ) RingHom ( 2nd𝑣 ) ) ↦ ( 𝑓𝑔 ) ) ) )
7 1 3 4 5 6 ringcvalALTV ( 𝜑𝐶 = { ⟨ ( Base ‘ ndx ) , ( 𝑈 ∩ Ring ) ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ ( 𝑈 ∩ Ring ) , 𝑦 ∈ ( 𝑈 ∩ Ring ) ↦ ( 𝑥 RingHom 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( ( 𝑈 ∩ Ring ) × ( 𝑈 ∩ Ring ) ) , 𝑧 ∈ ( 𝑈 ∩ Ring ) ↦ ( 𝑓 ∈ ( ( 2nd𝑣 ) RingHom 𝑧 ) , 𝑔 ∈ ( ( 1st𝑣 ) RingHom ( 2nd𝑣 ) ) ↦ ( 𝑓𝑔 ) ) ) ⟩ } )
8 catstr { ⟨ ( Base ‘ ndx ) , ( 𝑈 ∩ Ring ) ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ ( 𝑈 ∩ Ring ) , 𝑦 ∈ ( 𝑈 ∩ Ring ) ↦ ( 𝑥 RingHom 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( ( 𝑈 ∩ Ring ) × ( 𝑈 ∩ Ring ) ) , 𝑧 ∈ ( 𝑈 ∩ Ring ) ↦ ( 𝑓 ∈ ( ( 2nd𝑣 ) RingHom 𝑧 ) , 𝑔 ∈ ( ( 1st𝑣 ) RingHom ( 2nd𝑣 ) ) ↦ ( 𝑓𝑔 ) ) ) ⟩ } Struct ⟨ 1 , 1 5 ⟩
9 baseid Base = Slot ( Base ‘ ndx )
10 snsstp1 { ⟨ ( Base ‘ ndx ) , ( 𝑈 ∩ Ring ) ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , ( 𝑈 ∩ Ring ) ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ ( 𝑈 ∩ Ring ) , 𝑦 ∈ ( 𝑈 ∩ Ring ) ↦ ( 𝑥 RingHom 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( ( 𝑈 ∩ Ring ) × ( 𝑈 ∩ Ring ) ) , 𝑧 ∈ ( 𝑈 ∩ Ring ) ↦ ( 𝑓 ∈ ( ( 2nd𝑣 ) RingHom 𝑧 ) , 𝑔 ∈ ( ( 1st𝑣 ) RingHom ( 2nd𝑣 ) ) ↦ ( 𝑓𝑔 ) ) ) ⟩ }
11 inex1g ( 𝑈𝑉 → ( 𝑈 ∩ Ring ) ∈ V )
12 3 11 syl ( 𝜑 → ( 𝑈 ∩ Ring ) ∈ V )
13 7 8 9 10 12 2 strfv3 ( 𝜑𝐵 = ( 𝑈 ∩ Ring ) )