Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-rngiso Structured version   Visualization version   GIF version

Definition df-rngiso 18925
 Description: Define the set of ring isomorphisms from 𝑟 to 𝑠. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Assertion
Ref Expression
df-rngiso RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ 𝑓 ∈ (𝑠 RingHom 𝑟)})
Distinct variable group:   𝑠,𝑟,𝑓

Detailed syntax breakdown of Definition df-rngiso
StepHypRef Expression
1 crs 18922 . 2 class RingIso
2 vr . . 3 setvar 𝑟
3 vs . . 3 setvar 𝑠
4 cvv 3349 . . 3 class V
5 vf . . . . . . 7 setvar 𝑓
65cv 1629 . . . . . 6 class 𝑓
76ccnv 5248 . . . . 5 class 𝑓
83cv 1629 . . . . . 6 class 𝑠
92cv 1629 . . . . . 6 class 𝑟
10 crh 18921 . . . . . 6 class RingHom
118, 9, 10co 6792 . . . . 5 class (𝑠 RingHom 𝑟)
127, 11wcel 2144 . . . 4 wff 𝑓 ∈ (𝑠 RingHom 𝑟)
139, 8, 10co 6792 . . . 4 class (𝑟 RingHom 𝑠)
1412, 5, 13crab 3064 . . 3 class {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ 𝑓 ∈ (𝑠 RingHom 𝑟)}
152, 3, 4, 4, 14cmpt2 6794 . 2 class (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ 𝑓 ∈ (𝑠 RingHom 𝑟)})
161, 15wceq 1630 1 wff RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ 𝑓 ∈ (𝑠 RingHom 𝑟)})
 Colors of variables: wff setvar class This definition is referenced by:  isrim0  18932  rimrcl  18933  brric  18953
 Copyright terms: Public domain W3C validator