Theorem imasbas 12733

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 2177	. . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2177	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
5		eqid 2177	. . . 4 ⊢ ( ·𝑠 ‘𝑅) = ( ·𝑠 ‘𝑅)
6		eqidd 2178	. . . 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 2178	. . . 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 12732	. . 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 5519	. 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 12520	. . . . . 6 ⊢ (Base‘ndx) ∈ ℕ
13		basfn 12522	. . . . . . . . 9 ⊢ Base Fn V
14	9	elexd 2752	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ V)
15		funfvex 5534	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
16	15	funfni 5318	. . . . . . . . 9 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
17	13, 14, 16	sylancr 414	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
18	2, 17	eqeltrd 2254	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ V)
19		focdmex 6118	. . . . . . 7 ⊢ (𝑉 ∈ V → (𝐹:𝑉–onto→𝐵 → 𝐵 ∈ V))
20	18, 8, 19	sylc 62	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
21		opexg 4230	. . . . . 6 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝐵 ∈ V) → ⟨(Base‘ndx), 𝐵⟩ ∈ V)
22	12, 20, 21	sylancr 414	. . . . 5 ⊢ (𝜑 → ⟨(Base‘ndx), 𝐵⟩ ∈ V)
23		plusgndxnn 12572	. . . . . 6 ⊢ (+g‘ndx) ∈ ℕ
24		fof 5440	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
25	8, 24	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
26	25, 18	fexd 5748	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ V)
27		vex 2742	. . . . . . . . . . . . . 14 ⊢ 𝑝 ∈ V
28		fvexg 5536	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ V ∧ 𝑝 ∈ V) → (𝐹‘𝑝) ∈ V)
29	26, 27, 28	sylancl 413	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘𝑝) ∈ V)
30		vex 2742	. . . . . . . . . . . . . 14 ⊢ 𝑞 ∈ V
31		fvexg 5536	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ V ∧ 𝑞 ∈ V) → (𝐹‘𝑞) ∈ V)
32	26, 30, 31	sylancl 413	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘𝑞) ∈ V)
33		opexg 4230	. . . . . . . . . . . . 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 12573	. . . . . . . . . . . . . . . 16 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
37	36	slotex 12491	. . . . . . . . . . . . . . 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 5911	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ V ∧ (+g‘𝑅) ∈ V ∧ 𝑞 ∈ V) → (𝑝(+g‘𝑅)𝑞) ∈ V)
41	35, 38, 39, 40	syl3anc 1238	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑝(+g‘𝑅)𝑞) ∈ V)
42		fvexg 5536	. . . . . . . . . . . . 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 4230	. . . . . . . . . . . 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 4186	. . . . . . . . . . 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 2551	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
49		iunexg 6122	. . . . . . . . 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 2551	. . . . . . 7 ⊢ (𝜑 → ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
52		iunexg 6122	. . . . . . 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 4230	. . . . . 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 12592	. . . . . . 7 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
57	56	simpri 113	. . . . . 6 ⊢ (.r‘ndx) ∈ ℕ
58	56	slotex 12491	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ 𝑍 → (.r‘𝑅) ∈ V)
59	9, 58	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (.r‘𝑅) ∈ V)
60		ovexg 5911	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ V ∧ (.r‘𝑅) ∈ V ∧ 𝑞 ∈ V) → (𝑝(.r‘𝑅)𝑞) ∈ V)
61	35, 59, 39, 60	syl3anc 1238	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑝(.r‘𝑅)𝑞) ∈ V)
62		fvexg 5536	. . . . . . . . . . . . 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 4230	. . . . . . . . . . . 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 4186	. . . . . . . . . . 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 2551	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
69		iunexg 6122	. . . . . . . . 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 2551	. . . . . . 7 ⊢ (𝜑 → ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
72		iunexg 6122	. . . . . . 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 4230	. . . . . 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 4446	. . . . 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 1238	. . . 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 2254	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
79		baseid 12518	. . 3 ⊢ Base = Slot (Base‘ndx)
80	78, 79, 12	strndxid 12492	. 2 ⊢ (𝜑 → (𝑈‘(Base‘ndx)) = (Base‘𝑈))
81	12	a1i 9	. . 3 ⊢ (𝜑 → (Base‘ndx) ∈ ℕ)
82		basendxnplusgndx 12585	. . . 4 ⊢ (Base‘ndx) ≠ (+g‘ndx)
83	82	a1i 9	. . 3 ⊢ (𝜑 → (Base‘ndx) ≠ (+g‘ndx))
84		basendxnmulrndx 12594	. . . 4 ⊢ (Base‘ndx) ≠ (.r‘ndx)
85	84	a1i 9	. . 3 ⊢ (𝜑 → (Base‘ndx) ≠ (.r‘ndx))
86		fvtp1g 5726	. . 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 1239	. 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 2219	1 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148   ≠ wne 2347  ∀wral 2455  Vcvv 2739  {csn 3594  {ctp 3596  ⟨cop 3597  ∪ ciun 3888   Fn wfn 5213  ⟶wf 5214  –onto→wfo 5216  ‘cfv 5218  (class class class)co 5877  ℕcn 8921  ndxcnx 12461  Slot cslot 12463  Basecbs 12464  +_gcplusg 12538  ._rcmulr 12539   ·_𝑠 cvsca 12542   “_s cimas 12725

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-tp 3602  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-mulr 12552  df-iimas 12728

This theorem is referenced by:  qusbas  12752