Theorem imasex 13451

Description: Existence of the image structure. (Contributed by Jim Kingdon, 13-Mar-2025.)

Assertion

Ref	Expression
imasex	⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐹 “s 𝑅) ∈ V)

Proof of Theorem imasex

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

Step	Hyp	Ref	Expression
1		elex 2815	. . . 4 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2	1	adantr 276	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝐹 ∈ V)
3		elex 2815	. . . 4 ⊢ (𝑅 ∈ 𝑊 → 𝑅 ∈ V)
4	3	adantl 277	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑅 ∈ V)
5		basfn 13204	. . . . . 6 ⊢ Base Fn V
6		funfvex 5665	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
7	6	funfni 5439	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
8	5, 3, 7	sylancr 414	. . . . 5 ⊢ (𝑅 ∈ 𝑊 → (Base‘𝑅) ∈ V)
9	8	adantl 277	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (Base‘𝑅) ∈ V)
10		basendxnn 13201	. . . . . . 7 ⊢ (Base‘ndx) ∈ ℕ
11		rnexg 5003	. . . . . . . 8 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V)
12	11	adantr 276	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ran 𝐹 ∈ V)
13		opexg 4326	. . . . . . 7 ⊢ (((Base‘ndx) ∈ ℕ ∧ ran 𝐹 ∈ V) → ⟨(Base‘ndx), ran 𝐹⟩ ∈ V)
14	10, 12, 13	sylancr 414	. . . . . 6 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ⟨(Base‘ndx), ran 𝐹⟩ ∈ V)
15		plusgndxnn 13257	. . . . . . 7 ⊢ (+g‘ndx) ∈ ℕ
16		vex 2806	. . . . . . . 8 ⊢ 𝑣 ∈ V
17		vex 2806	. . . . . . . . . . . . . . . 16 ⊢ 𝑝 ∈ V
18	17	a1i 9	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑝 ∈ V)
19		fvexg 5667	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑝 ∈ V) → (𝐹‘𝑝) ∈ V)
20	18, 19	syldan 282	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐹‘𝑝) ∈ V)
21		vex 2806	. . . . . . . . . . . . . . . 16 ⊢ 𝑞 ∈ V
22	21	a1i 9	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → 𝑞 ∈ V)
23		fvexg 5667	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑞 ∈ V) → (𝐹‘𝑞) ∈ V)
24	22, 23	syldan 282	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐹‘𝑞) ∈ V)
25		opexg 4326	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑝) ∈ V ∧ (𝐹‘𝑞) ∈ V) → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ V)
26	20, 24, 25	syl2anc 411	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ V)
27		plusgslid 13258	. . . . . . . . . . . . . . . . 17 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
28	27	slotex 13172	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ 𝑊 → (+g‘𝑅) ∈ V)
29	28	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (+g‘𝑅) ∈ V)
30		ovexg 6062	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ V ∧ (+g‘𝑅) ∈ V ∧ 𝑞 ∈ V) → (𝑝(+g‘𝑅)𝑞) ∈ V)
31	18, 29, 22, 30	syl3anc 1274	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑝(+g‘𝑅)𝑞) ∈ V)
32		fvexg 5667	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ 𝑉 ∧ (𝑝(+g‘𝑅)𝑞) ∈ V) → (𝐹‘(𝑝(+g‘𝑅)𝑞)) ∈ V)
33	31, 32	syldan 282	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐹‘(𝑝(+g‘𝑅)𝑞)) ∈ V)
34		opexg 4326	. . . . . . . . . . . . 13 ⊢ ((⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ V ∧ (𝐹‘(𝑝(+g‘𝑅)𝑞)) ∈ V) → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩ ∈ V)
35	26, 33, 34	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩ ∈ V)
36		snexg 4280	. . . . . . . . . . . 12 ⊢ (⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩ ∈ V → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
37	35, 36	syl 14	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
38	37	ralrimivw 2607	. . . . . . . . . 10 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ∀𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
39		iunexg 6290	. . . . . . . . . 10 ⊢ ((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)
40	16, 38, 39	sylancr 414	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
41	40	ralrimivw 2607	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ∀𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
42		iunexg 6290	. . . . . . . 8 ⊢ ((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)
43	16, 41, 42	sylancr 414	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩} ∈ V)
44		opexg 4326	. . . . . . 7 ⊢ (((+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)
45	15, 43, 44	sylancr 414	. . . . . 6 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩ ∈ V)
46		mulrslid 13278	. . . . . . . 8 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
47	46	simpri 113	. . . . . . 7 ⊢ (.r‘ndx) ∈ ℕ
48	46	slotex 13172	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ 𝑊 → (.r‘𝑅) ∈ V)
49	48	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (.r‘𝑅) ∈ V)
50		ovexg 6062	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ V ∧ (.r‘𝑅) ∈ V ∧ 𝑞 ∈ V) → (𝑝(.r‘𝑅)𝑞) ∈ V)
51	18, 49, 22, 50	syl3anc 1274	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑝(.r‘𝑅)𝑞) ∈ V)
52		fvexg 5667	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ 𝑉 ∧ (𝑝(.r‘𝑅)𝑞) ∈ V) → (𝐹‘(𝑝(.r‘𝑅)𝑞)) ∈ V)
53	51, 52	syldan 282	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐹‘(𝑝(.r‘𝑅)𝑞)) ∈ V)
54		opexg 4326	. . . . . . . . . . . . 13 ⊢ ((⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ V ∧ (𝐹‘(𝑝(.r‘𝑅)𝑞)) ∈ V) → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩ ∈ V)
55	26, 53, 54	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩ ∈ V)
56		snexg 4280	. . . . . . . . . . . 12 ⊢ (⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩ ∈ V → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
57	55, 56	syl 14	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
58	57	ralrimivw 2607	. . . . . . . . . 10 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ∀𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
59		iunexg 6290	. . . . . . . . . 10 ⊢ ((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)
60	16, 58, 59	sylancr 414	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
61	60	ralrimivw 2607	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ∀𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
62		iunexg 6290	. . . . . . . 8 ⊢ ((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)
63	16, 61, 62	sylancr 414	. . . . . . 7 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩} ∈ V)
64		opexg 4326	. . . . . . 7 ⊢ (((.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)
65	47, 63, 64	sylancr 414	. . . . . 6 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩ ∈ V)
66		tpexg 4547	. . . . . 6 ⊢ ((⟨(Base‘ndx), ran 𝐹⟩ ∈ 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), 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))⟩}⟩} ∈ V)
67	14, 45, 65, 66	syl3anc 1274	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → {⟨(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))⟩}⟩} ∈ V)
68	67	alrimiv 1922	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ∀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))⟩}⟩} ∈ V)
69		csbexga 4222	. . . 4 ⊢ (((Base‘𝑅) ∈ V ∧ ∀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))⟩}⟩} ∈ 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))⟩}⟩} ∈ V)
70	9, 68, 69	syl2anc 411	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ⦋(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))⟩}⟩} ∈ V)
71		rneq 4965	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran 𝑓 = ran 𝐹)
72	71	opeq2d 3874	. . . . . 6 ⊢ (𝑓 = 𝐹 → ⟨(Base‘ndx), ran 𝑓⟩ = ⟨(Base‘ndx), ran 𝐹⟩)
73		fveq1 5647	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑝) = (𝐹‘𝑝))
74		fveq1 5647	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑞) = (𝐹‘𝑞))
75	73, 74	opeq12d 3875	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩ = ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩)
76		fveq1 5647	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑝(+g‘𝑟)𝑞)) = (𝐹‘(𝑝(+g‘𝑟)𝑞)))
77	75, 76	opeq12d 3875	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩ = ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑟)𝑞))⟩)
78	77	sneqd 3686	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩} = {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑟)𝑞))⟩})
79	78	iuneq2d 4000	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩} = ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑟)𝑞))⟩})
80	79	iuneq2d 4000	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ∪ 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))⟩})
81	80	opeq2d 3874	. . . . . 6 ⊢ (𝑓 = 𝐹 → ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(+g‘𝑟)𝑞))⟩}⟩ = ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑟)𝑞))⟩}⟩)
82		fveq1 5647	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑝(.r‘𝑟)𝑞)) = (𝐹‘(𝑝(.r‘𝑟)𝑞)))
83	75, 82	opeq12d 3875	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩ = ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑟)𝑞))⟩)
84	83	sneqd 3686	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩} = {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑟)𝑞))⟩})
85	84	iuneq2d 4000	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩} = ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑟)𝑞))⟩})
86	85	iuneq2d 4000	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ∪ 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))⟩})
87	86	opeq2d 3874	. . . . . 6 ⊢ (𝑓 = 𝐹 → ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝑓‘𝑝), (𝑓‘𝑞)⟩, (𝑓‘(𝑝(.r‘𝑟)𝑞))⟩}⟩ = ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑟)𝑞))⟩}⟩)
88	72, 81, 87	tpeq123d 3767	. . . . 5 ⊢ (𝑓 = 𝐹 → {⟨(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))⟩}⟩} = {⟨(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))⟩}⟩})
89	88	csbeq2dv 3154	. . . 4 ⊢ (𝑓 = 𝐹 → ⦋(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))⟩}⟩} = ⦋(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))⟩}⟩})
90		fveq2 5648	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
91	90	csbeq1d 3135	. . . . 5 ⊢ (𝑟 = 𝑅 → ⦋(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))⟩}⟩} = ⦋(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))⟩}⟩})
92		eqidd 2232	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ⟨(Base‘ndx), ran 𝐹⟩ = ⟨(Base‘ndx), ran 𝐹⟩)
93		fveq2 5648	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → (+g‘𝑟) = (+g‘𝑅))
94	93	oveqd 6045	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (𝑝(+g‘𝑟)𝑞) = (𝑝(+g‘𝑅)𝑞))
95	94	fveq2d 5652	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (𝐹‘(𝑝(+g‘𝑟)𝑞)) = (𝐹‘(𝑝(+g‘𝑅)𝑞)))
96	95	opeq2d 3874	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑟)𝑞))⟩ = ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩)
97	96	sneqd 3686	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑟)𝑞))⟩} = {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩})
98	97	iuneq2d 4000	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑟)𝑞))⟩} = ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩})
99	98	iuneq2d 4000	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ∪ 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))⟩})
100	99	opeq2d 3874	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑟)𝑞))⟩}⟩ = ⟨(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(+g‘𝑅)𝑞))⟩}⟩)
101		fveq2 5648	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
102	101	oveqd 6045	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (𝑝(.r‘𝑟)𝑞) = (𝑝(.r‘𝑅)𝑞))
103	102	fveq2d 5652	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (𝐹‘(𝑝(.r‘𝑟)𝑞)) = (𝐹‘(𝑝(.r‘𝑅)𝑞)))
104	103	opeq2d 3874	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑟)𝑞))⟩ = ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩)
105	104	sneqd 3686	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑟)𝑞))⟩} = {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩})
106	105	iuneq2d 4000	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑟)𝑞))⟩} = ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩})
107	106	iuneq2d 4000	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ∪ 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))⟩})
108	107	opeq2d 3874	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑟)𝑞))⟩}⟩ = ⟨(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝(.r‘𝑅)𝑞))⟩}⟩)
109	92, 100, 108	tpeq123d 3767	. . . . . 6 ⊢ (𝑟 = 𝑅 → {⟨(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))⟩}⟩} = {⟨(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))⟩}⟩})
110	109	csbeq2dv 3154	. . . . 5 ⊢ (𝑟 = 𝑅 → ⦋(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))⟩}⟩} = ⦋(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))⟩}⟩})
111	91, 110	eqtrd 2264	. . . 4 ⊢ (𝑟 = 𝑅 → ⦋(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))⟩}⟩} = ⦋(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))⟩}⟩})
112		df-iimas 13448	. . . 4 ⊢ “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))⟩}⟩})
113	89, 111, 112	ovmpog 6166	. . 3 ⊢ ((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))⟩}⟩} ∈ V) → (𝐹 “s 𝑅) = ⦋(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))⟩}⟩})
114	2, 4, 70, 113	syl3anc 1274	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐹 “s 𝑅) = ⦋(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))⟩}⟩})
115	114, 70	eqeltrd 2308	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐹 “s 𝑅) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1396   = wceq 1398   ∈ wcel 2202  ∀wral 2511  Vcvv 2803  ⦋csb 3128  {csn 3673  {ctp 3675  ⟨cop 3676  ∪ ciun 3975  ran crn 4732   Fn wfn 5328  ‘cfv 5333  (class class class)co 6028  ℕcn 9185  ndxcnx 13142  Slot cslot 13144  Basecbs 13145  +_gcplusg 13223  ._rcmulr 13224   “_s cimas 13445

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-inn 9186  df-2 9244  df-3 9245  df-ndx 13148  df-slot 13149  df-base 13151  df-plusg 13236  df-mulr 13237  df-iimas 13448

This theorem is referenced by:  imasmulr  13455  qusval  13469  qusex  13471  xpsval  13498