Definition df-rngim 20324

Description: Define the set of non-unital ring isomorphisms from 𝑟 to 𝑠. (Contributed by AV, 20-Feb-2020.)

Assertion

Ref	Expression
df-rngim	⊢ RngIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHom 𝑟)})

Distinct variable group:   𝑠,𝑟,𝑓

Detailed syntax breakdown of Definition df-rngim

Step	Hyp	Ref	Expression
1		crngim 20322	. 2 class RngIso
2		vr	. . 3 setvar 𝑟
3		vs	. . 3 setvar 𝑠
4		cvv 3466	. . 3 class V
5		vf	. . . . . . 7 setvar 𝑓
6	5	cv 1532	. . . . . 6 class 𝑓
7	6	ccnv 5665	. . . . 5 class ◡𝑓
8	3	cv 1532	. . . . . 6 class 𝑠
9	2	cv 1532	. . . . . 6 class 𝑟
10		crnghm 20321	. . . . . 6 class RngHom
11	8, 9, 10	co 7401	. . . . 5 class (𝑠 RngHom 𝑟)
12	7, 11	wcel 2098	. . . 4 wff ◡𝑓 ∈ (𝑠 RngHom 𝑟)
13	9, 8, 10	co 7401	. . . 4 class (𝑟 RngHom 𝑠)
14	12, 5, 13	crab 3424	. . 3 class {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHom 𝑟)}
15	2, 3, 4, 4, 14	cmpo 7403	. 2 class (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHom 𝑟)})
16	1, 15	wceq 1533	1 wff RngIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHom 𝑟)})

This definition is referenced by:  isrngim  20332  rngimrcl  20333