Step | Hyp | Ref
| Expression |
1 | | csf 34618 |
. 2
class
splitFld |
2 | | vr |
. . 3
setvar 𝑟 |
3 | | vp |
. . 3
setvar 𝑝 |
4 | | cvv 3474 |
. . 3
class
V |
5 | | c1 11110 |
. . . . . . . 8
class
1 |
6 | 3 | cv 1540 |
. . . . . . . . 9
class 𝑝 |
7 | | chash 14289 |
. . . . . . . . 9
class
♯ |
8 | 6, 7 | cfv 6543 |
. . . . . . . 8
class
(♯‘𝑝) |
9 | | cfz 13483 |
. . . . . . . 8
class
... |
10 | 5, 8, 9 | co 7408 |
. . . . . . 7
class
(1...(♯‘𝑝)) |
11 | | clt 11247 |
. . . . . . 7
class
< |
12 | 2 | cv 1540 |
. . . . . . . 8
class 𝑟 |
13 | | cplt 18260 |
. . . . . . . 8
class
lt |
14 | 12, 13 | cfv 6543 |
. . . . . . 7
class
(lt‘𝑟) |
15 | | vf |
. . . . . . . 8
setvar 𝑓 |
16 | 15 | cv 1540 |
. . . . . . 7
class 𝑓 |
17 | 10, 6, 11, 14, 16 | wiso 6544 |
. . . . . 6
wff 𝑓 Isom < , (lt‘𝑟)((1...(♯‘𝑝)), 𝑝) |
18 | | vx |
. . . . . . . 8
setvar 𝑥 |
19 | 18 | cv 1540 |
. . . . . . 7
class 𝑥 |
20 | | ve |
. . . . . . . . . 10
setvar 𝑒 |
21 | | vg |
. . . . . . . . . 10
setvar 𝑔 |
22 | 21 | cv 1540 |
. . . . . . . . . . 11
class 𝑔 |
23 | 20 | cv 1540 |
. . . . . . . . . . . 12
class 𝑒 |
24 | | csf1 34617 |
. . . . . . . . . . . 12
class
splitFld1 |
25 | 12, 23, 24 | co 7408 |
. . . . . . . . . . 11
class (𝑟 splitFld1 𝑒) |
26 | 22, 25 | cfv 6543 |
. . . . . . . . . 10
class ((𝑟 splitFld1 𝑒)‘𝑔) |
27 | 20, 21, 4, 4, 26 | cmpo 7410 |
. . . . . . . . 9
class (𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)) |
28 | | cc0 11109 |
. . . . . . . . . . . 12
class
0 |
29 | | cid 5573 |
. . . . . . . . . . . . . 14
class
I |
30 | | cbs 17143 |
. . . . . . . . . . . . . . 15
class
Base |
31 | 12, 30 | cfv 6543 |
. . . . . . . . . . . . . 14
class
(Base‘𝑟) |
32 | 29, 31 | cres 5678 |
. . . . . . . . . . . . 13
class ( I
↾ (Base‘𝑟)) |
33 | 12, 32 | cop 4634 |
. . . . . . . . . . . 12
class
⟨𝑟, ( I ↾
(Base‘𝑟))⟩ |
34 | 28, 33 | cop 4634 |
. . . . . . . . . . 11
class ⟨0,
⟨𝑟, ( I ↾
(Base‘𝑟))⟩⟩ |
35 | 34 | csn 4628 |
. . . . . . . . . 10
class {⟨0,
⟨𝑟, ( I ↾
(Base‘𝑟))⟩⟩} |
36 | 16, 35 | cun 3946 |
. . . . . . . . 9
class (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}) |
37 | 27, 36, 28 | cseq 13965 |
. . . . . . . 8
class
seq0((𝑒 ∈ V,
𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩})) |
38 | 8, 37 | cfv 6543 |
. . . . . . 7
class
(seq0((𝑒 ∈ V,
𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(♯‘𝑝)) |
39 | 19, 38 | wceq 1541 |
. . . . . 6
wff 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(♯‘𝑝)) |
40 | 17, 39 | wa 396 |
. . . . 5
wff (𝑓 Isom < , (lt‘𝑟)((1...(♯‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(♯‘𝑝))) |
41 | 40, 15 | wex 1781 |
. . . 4
wff
∃𝑓(𝑓 Isom < , (lt‘𝑟)((1...(♯‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(♯‘𝑝))) |
42 | 41, 18 | cio 6493 |
. . 3
class
(℩𝑥∃𝑓(𝑓 Isom < , (lt‘𝑟)((1...(♯‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(♯‘𝑝)))) |
43 | 2, 3, 4, 4, 42 | cmpo 7410 |
. 2
class (𝑟 ∈ V, 𝑝 ∈ V ↦ (℩𝑥∃𝑓(𝑓 Isom < , (lt‘𝑟)((1...(♯‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(♯‘𝑝))))) |
44 | 1, 43 | wceq 1541 |
1
wff splitFld =
(𝑟 ∈ V, 𝑝 ∈ V ↦ (℩𝑥∃𝑓(𝑓 Isom < , (lt‘𝑟)((1...(♯‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(♯‘𝑝))))) |