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 ( 𝜑𝑈𝑉 )
rngchomfvalALTV.h 𝐻 = ( Hom ‘ 𝐶 )
Assertion rngchomfvalALTV ( 𝜑𝐻 = ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥 RngHomo 𝑦 ) ) )

Proof

Step Hyp Ref Expression
1 rngcbasALTV.c 𝐶 = ( RngCatALTV ‘ 𝑈 )
2 rngcbasALTV.b 𝐵 = ( Base ‘ 𝐶 )
3 rngcbasALTV.u ( 𝜑𝑈𝑉 )
4 rngchomfvalALTV.h 𝐻 = ( Hom ‘ 𝐶 )
5 1 2 3 rngcbasALTV ( 𝜑𝐵 = ( 𝑈 ∩ Rng ) )
6 eqidd ( 𝜑 → ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥 RngHomo 𝑦 ) ) = ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥 RngHomo 𝑦 ) ) )
7 eqidd ( 𝜑 → ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑓 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑔 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑓𝑔 ) ) ) = ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑓 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑔 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑓𝑔 ) ) ) )
8 1 3 5 6 7 rngcvalALTV ( 𝜑𝐶 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥 RngHomo 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑓 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑔 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑓𝑔 ) ) ) ⟩ } )
9 8 fveq2d ( 𝜑 → ( Hom ‘ 𝐶 ) = ( Hom ‘ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥 RngHomo 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑓 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑔 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑓𝑔 ) ) ) ⟩ } ) )
10 4 9 syl5eq ( 𝜑𝐻 = ( Hom ‘ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥 RngHomo 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑓 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑔 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑓𝑔 ) ) ) ⟩ } ) )
11 2 fvexi 𝐵 ∈ V
12 11 11 mpoex ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥 RngHomo 𝑦 ) ) ∈ V
13 catstr { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥 RngHomo 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑓 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑔 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑓𝑔 ) ) ) ⟩ } Struct ⟨ 1 , 1 5 ⟩
14 homid Hom = Slot ( Hom ‘ ndx )
15 snsstp2 { ⟨ ( Hom ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥 RngHomo 𝑦 ) ) ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥 RngHomo 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑓 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑔 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑓𝑔 ) ) ) ⟩ }
16 13 14 15 strfv ( ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥 RngHomo 𝑦 ) ) ∈ V → ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥 RngHomo 𝑦 ) ) = ( Hom ‘ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥 RngHomo 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑓 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑔 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑓𝑔 ) ) ) ⟩ } ) )
17 12 16 mp1i ( 𝜑 → ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥 RngHomo 𝑦 ) ) = ( Hom ‘ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥 RngHomo 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧𝐵 ↦ ( 𝑓 ∈ ( ( 2nd𝑣 ) RngHomo 𝑧 ) , 𝑔 ∈ ( ( 1st𝑣 ) RngHomo ( 2nd𝑣 ) ) ↦ ( 𝑓𝑔 ) ) ) ⟩ } ) )
18 10 17 eqtr4d ( 𝜑𝐻 = ( 𝑥𝐵 , 𝑦𝐵 ↦ ( 𝑥 RngHomo 𝑦 ) ) )