Metamath Proof Explorer


Theorem rnghmfn

Description: The mapping of two non-unital rings to the non-unital ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020)

Ref Expression
Assertion rnghmfn RngHomo Fn ( Rng × Rng )

Proof

Step Hyp Ref Expression
1 df-rnghomo RngHomo = ( 𝑟 ∈ Rng , 𝑠 ∈ Rng ↦ ( Base ‘ 𝑟 ) / 𝑣 ( Base ‘ 𝑠 ) / 𝑤 { 𝑓 ∈ ( 𝑤m 𝑣 ) ∣ ∀ 𝑥𝑣𝑦𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g𝑟 ) 𝑦 ) ) = ( ( 𝑓𝑥 ) ( +g𝑠 ) ( 𝑓𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r𝑟 ) 𝑦 ) ) = ( ( 𝑓𝑥 ) ( .r𝑠 ) ( 𝑓𝑦 ) ) ) } )
2 ovex ( 𝑤m 𝑣 ) ∈ V
3 2 rabex { 𝑓 ∈ ( 𝑤m 𝑣 ) ∣ ∀ 𝑥𝑣𝑦𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g𝑟 ) 𝑦 ) ) = ( ( 𝑓𝑥 ) ( +g𝑠 ) ( 𝑓𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r𝑟 ) 𝑦 ) ) = ( ( 𝑓𝑥 ) ( .r𝑠 ) ( 𝑓𝑦 ) ) ) } ∈ V
4 3 csbex ( Base ‘ 𝑠 ) / 𝑤 { 𝑓 ∈ ( 𝑤m 𝑣 ) ∣ ∀ 𝑥𝑣𝑦𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g𝑟 ) 𝑦 ) ) = ( ( 𝑓𝑥 ) ( +g𝑠 ) ( 𝑓𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r𝑟 ) 𝑦 ) ) = ( ( 𝑓𝑥 ) ( .r𝑠 ) ( 𝑓𝑦 ) ) ) } ∈ V
5 4 csbex ( Base ‘ 𝑟 ) / 𝑣 ( Base ‘ 𝑠 ) / 𝑤 { 𝑓 ∈ ( 𝑤m 𝑣 ) ∣ ∀ 𝑥𝑣𝑦𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g𝑟 ) 𝑦 ) ) = ( ( 𝑓𝑥 ) ( +g𝑠 ) ( 𝑓𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r𝑟 ) 𝑦 ) ) = ( ( 𝑓𝑥 ) ( .r𝑠 ) ( 𝑓𝑦 ) ) ) } ∈ V
6 1 5 fnmpoi RngHomo Fn ( Rng × Rng )