Theorem imasbas 13009

Description: The base set of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 6-Oct-2020.)

Hypotheses

Ref	Expression
imasbas.u	⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
imasbas.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasbas.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imasbas.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)

Assertion

Ref	Expression
imasbas	⊢ (𝜑 → 𝐵 = (Base‘𝑈))

Proof of Theorem imasbas

Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasbas.u	. . . 4 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
2		imasbas.v	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		eqid 2196	. . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2196	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
5		eqid 2196	. . . 4 ⊢ ( ·𝑠 ‘𝑅) = ( ·𝑠 ‘𝑅)
6		eqidd 2197	. . . 4 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩})
7		eqidd 2197	. . . 4 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩})
8		imasbas.f	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
9		imasbas.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	imasival 13008	. . 3 ⊢ (𝜑 → 𝑈 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+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))⟩}⟩})
11	10	fveq1d 5563	. 2 ⊢ (𝜑 → (𝑈‘(Base‘ndx)) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+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))⟩}⟩}‘(Base‘ndx)))
12		basendxnn 12759	. . . . . 6 ⊢ (Base‘ndx) ∈ ℕ
13		basfn 12761	. . . . . . . . 9 ⊢ Base Fn V
14	9	elexd 2776	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ V)
15		funfvex 5578	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
16	15	funfni 5361	. . . . . . . . 9 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
17	13, 14, 16	sylancr 414	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
18	2, 17	eqeltrd 2273	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ V)
19		focdmex 6181	. . . . . . 7 ⊢ (𝑉 ∈ V → (𝐹:𝑉–onto→𝐵 → 𝐵 ∈ V))
20	18, 8, 19	sylc 62	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
21		opexg 4262	. . . . . 6 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝐵 ∈ V) → ⟨(Base‘ndx), 𝐵⟩ ∈ V)
22	12, 20, 21	sylancr 414	. . . . 5 ⊢ (𝜑 → ⟨(Base‘ndx), 𝐵⟩ ∈ V)
23		plusgndxnn 12814	. . . . . 6 ⊢ (+g‘ndx) ∈ ℕ
24		fof 5483	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
25	8, 24	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
26	25, 18	fexd 5795	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ V)
27		vex 2766	. . . . . . . . . . . . . 14 ⊢ 𝑝 ∈ V
28		fvexg 5580	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ V ∧ 𝑝 ∈ V) → (𝐹‘𝑝) ∈ V)
29	26, 27, 28	sylancl 413	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘𝑝) ∈ V)
30		vex 2766	. . . . . . . . . . . . . 14 ⊢ 𝑞 ∈ V
31		fvexg 5580	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ V ∧ 𝑞 ∈ V) → (𝐹‘𝑞) ∈ V)
32	26, 30, 31	sylancl 413	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘𝑞) ∈ V)
33		opexg 4262	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑝) ∈ V ∧ (𝐹‘𝑞) ∈ V) → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ V)
34	29, 32, 33	syl2anc 411	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ V)
35	27	a1i 9	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑝 ∈ V)
36		plusgslid 12815	. . . . . . . . . . . . . . . 16 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
37	36	slotex 12730	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ 𝑍 → (+g‘𝑅) ∈ V)
38	9, 37	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (+g‘𝑅) ∈ V)
39	30	a1i 9	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑞 ∈ V)
40		ovexg 5959	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ V ∧ (+g‘𝑅) ∈ V ∧ 𝑞 ∈ V) → (𝑝(+g‘𝑅)𝑞) ∈ V)
41	35, 38, 39, 40	syl3anc 1249	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑝(+g‘𝑅)𝑞) ∈ V)
42		fvexg 5580	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ V ∧ (𝑝(+g‘𝑅)𝑞) ∈ V) → (𝐹‘(𝑝(+g‘𝑅)𝑞)) ∈ V)
43	26, 41, 42	syl2anc 411	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘(𝑝(+g‘𝑅)𝑞)) ∈ V)
44		opexg 4262	. . . . . . . . . . . 12 ⊢ ((⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ V ∧ (𝐹‘(𝑝(+g‘𝑅)𝑞)) ∈ V) → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩ ∈ V)
45	34, 43, 44	syl2anc 411	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩ ∈ V)
46		snexg 4218	. . . . . . . . . . 11 ⊢ (⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩ ∈ V → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
47	45, 46	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
48	47	ralrimivw 2571	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
49		iunexg 6185	. . . . . . . . 9 ⊢ ((𝑉 ∈ V ∧ ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V) → ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
50	18, 48, 49	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
51	50	ralrimivw 2571	. . . . . . 7 ⊢ (𝜑 → ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
52		iunexg 6185	. . . . . . 7 ⊢ ((𝑉 ∈ V ∧ ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V) → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
53	18, 51, 52	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
54		opexg 4262	. . . . . 6 ⊢ (((+g‘ndx) ∈ ℕ ∧ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V) → ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩ ∈ V)
55	23, 53, 54	sylancr 414	. . . . 5 ⊢ (𝜑 → ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩ ∈ V)
56		mulrslid 12834	. . . . . . 7 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
57	56	simpri 113	. . . . . 6 ⊢ (.r‘ndx) ∈ ℕ
58	56	slotex 12730	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ 𝑍 → (.r‘𝑅) ∈ V)
59	9, 58	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (.r‘𝑅) ∈ V)
60		ovexg 5959	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ V ∧ (.r‘𝑅) ∈ V ∧ 𝑞 ∈ V) → (𝑝(.r‘𝑅)𝑞) ∈ V)
61	35, 59, 39, 60	syl3anc 1249	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑝(.r‘𝑅)𝑞) ∈ V)
62		fvexg 5580	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ V ∧ (𝑝(.r‘𝑅)𝑞) ∈ V) → (𝐹‘(𝑝(.r‘𝑅)𝑞)) ∈ V)
63	26, 61, 62	syl2anc 411	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘(𝑝(.r‘𝑅)𝑞)) ∈ V)
64		opexg 4262	. . . . . . . . . . . 12 ⊢ ((⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ V ∧ (𝐹‘(𝑝(.r‘𝑅)𝑞)) ∈ V) → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩ ∈ V)
65	34, 63, 64	syl2anc 411	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩ ∈ V)
66		snexg 4218	. . . . . . . . . . 11 ⊢ (⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩ ∈ V → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
67	65, 66	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
68	67	ralrimivw 2571	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
69		iunexg 6185	. . . . . . . . 9 ⊢ ((𝑉 ∈ V ∧ ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V) → ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
70	18, 68, 69	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
71	70	ralrimivw 2571	. . . . . . 7 ⊢ (𝜑 → ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
72		iunexg 6185	. . . . . . 7 ⊢ ((𝑉 ∈ V ∧ ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V) → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
73	18, 71, 72	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
74		opexg 4262	. . . . . 6 ⊢ (((.r‘ndx) ∈ ℕ ∧ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V) → ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩ ∈ V)
75	57, 73, 74	sylancr 414	. . . . 5 ⊢ (𝜑 → ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩ ∈ V)
76		tpexg 4480	. . . . 5 ⊢ ((⟨(Base‘ndx), 𝐵⟩ ∈ V ∧ ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩ ∈ V ∧ ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩ ∈ V) → {⟨(Base‘ndx), 𝐵⟩, ⟨(+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))⟩}⟩} ∈ V)
77	22, 55, 75, 76	syl3anc 1249	. . . 4 ⊢ (𝜑 → {⟨(Base‘ndx), 𝐵⟩, ⟨(+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))⟩}⟩} ∈ V)
78	10, 77	eqeltrd 2273	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
79		baseid 12757	. . 3 ⊢ Base = Slot (Base‘ndx)
80	78, 79, 12	strndxid 12731	. 2 ⊢ (𝜑 → (𝑈‘(Base‘ndx)) = (Base‘𝑈))
81	12	a1i 9	. . 3 ⊢ (𝜑 → (Base‘ndx) ∈ ℕ)
82		basendxnplusgndx 12827	. . . 4 ⊢ (Base‘ndx) ≠ (+g‘ndx)
83	82	a1i 9	. . 3 ⊢ (𝜑 → (Base‘ndx) ≠ (+g‘ndx))
84		basendxnmulrndx 12836	. . . 4 ⊢ (Base‘ndx) ≠ (.r‘ndx)
85	84	a1i 9	. . 3 ⊢ (𝜑 → (Base‘ndx) ≠ (.r‘ndx))
86		fvtp1g 5773	. . 3 ⊢ ((((Base‘ndx) ∈ ℕ ∧ 𝐵 ∈ V) ∧ ((Base‘ndx) ≠ (+g‘ndx) ∧ (Base‘ndx) ≠ (.r‘ndx))) → ({⟨(Base‘ndx), 𝐵⟩, ⟨(+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))⟩}⟩}‘(Base‘ndx)) = 𝐵)
87	81, 20, 83, 85, 86	syl22anc 1250	. 2 ⊢ (𝜑 → ({⟨(Base‘ndx), 𝐵⟩, ⟨(+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))⟩}⟩}‘(Base‘ndx)) = 𝐵)
88	11, 80, 87	3eqtr3rd 2238	1 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167   ≠ wne 2367  ∀wral 2475  Vcvv 2763  {csn 3623  {ctp 3625  ⟨cop 3626  ∪ ciun 3917   Fn wfn 5254  ⟶wf 5255  –onto→wfo 5257  ‘cfv 5259  (class class class)co 5925  ℕcn 9007  ndxcnx 12700  Slot cslot 12702  Basecbs 12703  +_gcplusg 12780  ._rcmulr 12781   ·_𝑠 cvsca 12784   “_s cimas 13001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-tp 3631  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-mulr 12794  df-iimas 13004

This theorem is referenced by:  qusbas  13029  imasmnd2  13154  imasgrp2  13316  imasabl  13542  imasrng  13588  imasring  13696