Definition df-iimas 12722

Description: Define an image structure, which takes a structure and a function on the base set, and maps all the operations via the function. For this to work properly 𝑓 must either be injective or satisfy the well-definedness condition 𝑓(𝑎) = 𝑓(𝑐) ∧ 𝑓(𝑏) = 𝑓(𝑑) → 𝑓(𝑎 + 𝑏) = 𝑓(𝑐 + 𝑑) for each relevant operation.

Note that although we call this an "image" by association to df-ima 4639, in order to keep the definition simple we consider only the case when the domain of 𝐹 is equal to the base set of 𝑅. Other cases can be achieved by restricting 𝐹 (with df-res 4638) and/or 𝑅 ( with df-iress 12469) to their common domain. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by AV, 6-Oct-2020.)

Assertion

Ref	Expression
df-iimas	⊢ “s = (𝑓 ∈ V, 𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑣⦌{⟨(Base‘ndx), ran 𝑓⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩}⟩})

Distinct variable group:   𝑓,𝑝,𝑞,𝑟,𝑣

Detailed syntax breakdown of Definition df-iimas

Step	Hyp	Ref	Expression
1		cimas 12719	. 2 class “s
2		vf	. . 3 setvar 𝑓
3		vr	. . 3 setvar 𝑟
4		cvv 2737	. . 3 class V
5		vv	. . . 4 setvar 𝑣
6	3	cv 1352	. . . . 5 class 𝑟
7		cbs 12461	. . . . 5 class Base
8	6, 7	cfv 5216	. . . 4 class (Base‘𝑟)
9		cnx 12458	. . . . . . 7 class ndx
10	9, 7	cfv 5216	. . . . . 6 class (Base‘ndx)
11	2	cv 1352	. . . . . . 7 class 𝑓
12	11	crn 4627	. . . . . 6 class ran 𝑓
13	10, 12	cop 3595	. . . . 5 class ⟨(Base‘ndx), ran 𝑓⟩
14		cplusg 12535	. . . . . . 7 class +g
15	9, 14	cfv 5216	. . . . . 6 class (+g‘ndx)
16		vp	. . . . . . 7 setvar 𝑝
17	5	cv 1352	. . . . . . 7 class 𝑣
18		vq	. . . . . . . 8 setvar 𝑞
19	16	cv 1352	. . . . . . . . . . . 12 class 𝑝
20	19, 11	cfv 5216	. . . . . . . . . . 11 class (𝑓‘𝑝)
21	18	cv 1352	. . . . . . . . . . . 12 class 𝑞
22	21, 11	cfv 5216	. . . . . . . . . . 11 class (𝑓‘𝑞)
23	20, 22	cop 3595	. . . . . . . . . 10 class ⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩
24	6, 14	cfv 5216	. . . . . . . . . . . 12 class (+g‘𝑟)
25	19, 21, 24	co 5874	. . . . . . . . . . 11 class (𝑝(+g‘𝑟)𝑞)
26	25, 11	cfv 5216	. . . . . . . . . 10 class (𝑓‘(𝑝(+g‘𝑟)𝑞))
27	23, 26	cop 3595	. . . . . . . . 9 class ⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩
28	27	csn 3592	. . . . . . . 8 class {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩}
29	18, 17, 28	ciun 3886	. . . . . . 7 class ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩}
30	16, 17, 29	ciun 3886	. . . . . 6 class ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩}
31	15, 30	cop 3595	. . . . 5 class ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩}⟩
32		cmulr 12536	. . . . . . 7 class .r
33	9, 32	cfv 5216	. . . . . 6 class (.r‘ndx)
34	6, 32	cfv 5216	. . . . . . . . . . . 12 class (.r‘𝑟)
35	19, 21, 34	co 5874	. . . . . . . . . . 11 class (𝑝(.r‘𝑟)𝑞)
36	35, 11	cfv 5216	. . . . . . . . . 10 class (𝑓‘(𝑝(.r‘𝑟)𝑞))
37	23, 36	cop 3595	. . . . . . . . 9 class ⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩
38	37	csn 3592	. . . . . . . 8 class {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩}
39	18, 17, 38	ciun 3886	. . . . . . 7 class ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩}
40	16, 17, 39	ciun 3886	. . . . . 6 class ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩}
41	33, 40	cop 3595	. . . . 5 class ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩}⟩
42	13, 31, 41	ctp 3594	. . . 4 class {⟨(Base‘ndx), ran 𝑓⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩}⟩}
43	5, 8, 42	csb 3057	. . 3 class ⦋(Base‘𝑟) / 𝑣⦌{⟨(Base‘ndx), ran 𝑓⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩}⟩}
44	2, 3, 4, 4, 43	cmpo 5876	. 2 class (𝑓 ∈ V, 𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑣⦌{⟨(Base‘ndx), ran 𝑓⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩}⟩})
45	1, 44	wceq 1353	1 wff “s = (𝑓 ∈ V, 𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑣⦌{⟨(Base‘ndx), ran 𝑓⟩, ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩}⟩, ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩}⟩})

This definition is referenced by:  imasex  12725  imasival  12726