Step | Hyp | Ref
| Expression |
1 | | copws 21461 |
. 2
class
ordPwSer |
2 | | vi |
. . 3
setvar π |
3 | | vs |
. . 3
setvar π |
4 | | cvv 3475 |
. . 3
class
V |
5 | | vr |
. . . 4
setvar π |
6 | 2 | cv 1541 |
. . . . . 6
class π |
7 | 6, 6 | cxp 5675 |
. . . . 5
class (π Γ π) |
8 | 7 | cpw 4603 |
. . . 4
class π«
(π Γ π) |
9 | | vp |
. . . . 5
setvar π |
10 | 3 | cv 1541 |
. . . . . 6
class π |
11 | | cmps 21457 |
. . . . . 6
class
mPwSer |
12 | 6, 10, 11 | co 7409 |
. . . . 5
class (π mPwSer π ) |
13 | 9 | cv 1541 |
. . . . . 6
class π |
14 | | cnx 17126 |
. . . . . . . 8
class
ndx |
15 | | cple 17204 |
. . . . . . . 8
class
le |
16 | 14, 15 | cfv 6544 |
. . . . . . 7
class
(leβndx) |
17 | | vx |
. . . . . . . . . . . 12
setvar π₯ |
18 | 17 | cv 1541 |
. . . . . . . . . . 11
class π₯ |
19 | | vy |
. . . . . . . . . . . 12
setvar π¦ |
20 | 19 | cv 1541 |
. . . . . . . . . . 11
class π¦ |
21 | 18, 20 | cpr 4631 |
. . . . . . . . . 10
class {π₯, π¦} |
22 | | cbs 17144 |
. . . . . . . . . . 11
class
Base |
23 | 13, 22 | cfv 6544 |
. . . . . . . . . 10
class
(Baseβπ) |
24 | 21, 23 | wss 3949 |
. . . . . . . . 9
wff {π₯, π¦} β (Baseβπ) |
25 | | vz |
. . . . . . . . . . . . . . . 16
setvar π§ |
26 | 25 | cv 1541 |
. . . . . . . . . . . . . . 15
class π§ |
27 | 26, 18 | cfv 6544 |
. . . . . . . . . . . . . 14
class (π₯βπ§) |
28 | 26, 20 | cfv 6544 |
. . . . . . . . . . . . . 14
class (π¦βπ§) |
29 | | cplt 18261 |
. . . . . . . . . . . . . . 15
class
lt |
30 | 10, 29 | cfv 6544 |
. . . . . . . . . . . . . 14
class
(ltβπ ) |
31 | 27, 28, 30 | wbr 5149 |
. . . . . . . . . . . . 13
wff (π₯βπ§)(ltβπ )(π¦βπ§) |
32 | | vw |
. . . . . . . . . . . . . . . . 17
setvar π€ |
33 | 32 | cv 1541 |
. . . . . . . . . . . . . . . 16
class π€ |
34 | 5 | cv 1541 |
. . . . . . . . . . . . . . . . 17
class π |
35 | | cltb 21460 |
. . . . . . . . . . . . . . . . 17
class
<bag |
36 | 34, 6, 35 | co 7409 |
. . . . . . . . . . . . . . . 16
class (π <bag π) |
37 | 33, 26, 36 | wbr 5149 |
. . . . . . . . . . . . . . 15
wff π€(π <bag π)π§ |
38 | 33, 18 | cfv 6544 |
. . . . . . . . . . . . . . . 16
class (π₯βπ€) |
39 | 33, 20 | cfv 6544 |
. . . . . . . . . . . . . . . 16
class (π¦βπ€) |
40 | 38, 39 | wceq 1542 |
. . . . . . . . . . . . . . 15
wff (π₯βπ€) = (π¦βπ€) |
41 | 37, 40 | wi 4 |
. . . . . . . . . . . . . 14
wff (π€(π <bag π)π§ β (π₯βπ€) = (π¦βπ€)) |
42 | | vd |
. . . . . . . . . . . . . . 15
setvar π |
43 | 42 | cv 1541 |
. . . . . . . . . . . . . 14
class π |
44 | 41, 32, 43 | wral 3062 |
. . . . . . . . . . . . 13
wff
βπ€ β
π (π€(π <bag π)π§ β (π₯βπ€) = (π¦βπ€)) |
45 | 31, 44 | wa 397 |
. . . . . . . . . . . 12
wff ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β π (π€(π <bag π)π§ β (π₯βπ€) = (π¦βπ€))) |
46 | 45, 25, 43 | wrex 3071 |
. . . . . . . . . . 11
wff
βπ§ β
π ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β π (π€(π <bag π)π§ β (π₯βπ€) = (π¦βπ€))) |
47 | | vh |
. . . . . . . . . . . . . . . 16
setvar β |
48 | 47 | cv 1541 |
. . . . . . . . . . . . . . 15
class β |
49 | 48 | ccnv 5676 |
. . . . . . . . . . . . . 14
class β‘β |
50 | | cn 12212 |
. . . . . . . . . . . . . 14
class
β |
51 | 49, 50 | cima 5680 |
. . . . . . . . . . . . 13
class (β‘β β β) |
52 | | cfn 8939 |
. . . . . . . . . . . . 13
class
Fin |
53 | 51, 52 | wcel 2107 |
. . . . . . . . . . . 12
wff (β‘β β β) β Fin |
54 | | cn0 12472 |
. . . . . . . . . . . . 13
class
β0 |
55 | | cmap 8820 |
. . . . . . . . . . . . 13
class
βm |
56 | 54, 6, 55 | co 7409 |
. . . . . . . . . . . 12
class
(β0 βm π) |
57 | 53, 47, 56 | crab 3433 |
. . . . . . . . . . 11
class {β β (β0
βm π)
β£ (β‘β β β) β Fin} |
58 | 46, 42, 57 | wsbc 3778 |
. . . . . . . . . 10
wff
[{β β
(β0 βm π) β£ (β‘β β β) β Fin} / π]βπ§ β π ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β π (π€(π <bag π)π§ β (π₯βπ€) = (π¦βπ€))) |
59 | 17, 19 | weq 1967 |
. . . . . . . . . 10
wff π₯ = π¦ |
60 | 58, 59 | wo 846 |
. . . . . . . . 9
wff
([{β β
(β0 βm π) β£ (β‘β β β) β Fin} / π]βπ§ β π ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β π (π€(π <bag π)π§ β (π₯βπ€) = (π¦βπ€))) β¨ π₯ = π¦) |
61 | 24, 60 | wa 397 |
. . . . . . . 8
wff ({π₯, π¦} β (Baseβπ) β§ ([{β β (β0
βm π)
β£ (β‘β β β) β Fin} / π]βπ§ β π ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β π (π€(π <bag π)π§ β (π₯βπ€) = (π¦βπ€))) β¨ π₯ = π¦)) |
62 | 61, 17, 19 | copab 5211 |
. . . . . . 7
class
{β¨π₯, π¦β© β£ ({π₯, π¦} β (Baseβπ) β§ ([{β β (β0
βm π)
β£ (β‘β β β) β Fin} / π]βπ§ β π ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β π (π€(π <bag π)π§ β (π₯βπ€) = (π¦βπ€))) β¨ π₯ = π¦))} |
63 | 16, 62 | cop 4635 |
. . . . . 6
class
β¨(leβndx), {β¨π₯, π¦β© β£ ({π₯, π¦} β (Baseβπ) β§ ([{β β (β0
βm π)
β£ (β‘β β β) β Fin} / π]βπ§ β π ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β π (π€(π <bag π)π§ β (π₯βπ€) = (π¦βπ€))) β¨ π₯ = π¦))}β© |
64 | | csts 17096 |
. . . . . 6
class
sSet |
65 | 13, 63, 64 | co 7409 |
. . . . 5
class (π sSet β¨(leβndx),
{β¨π₯, π¦β© β£ ({π₯, π¦} β (Baseβπ) β§ ([{β β (β0
βm π)
β£ (β‘β β β) β Fin} / π]βπ§ β π ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β π (π€(π <bag π)π§ β (π₯βπ€) = (π¦βπ€))) β¨ π₯ = π¦))}β©) |
66 | 9, 12, 65 | csb 3894 |
. . . 4
class
β¦(π
mPwSer π ) / πβ¦(π sSet β¨(leβndx),
{β¨π₯, π¦β© β£ ({π₯, π¦} β (Baseβπ) β§ ([{β β (β0
βm π)
β£ (β‘β β β) β Fin} / π]βπ§ β π ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β π (π€(π <bag π)π§ β (π₯βπ€) = (π¦βπ€))) β¨ π₯ = π¦))}β©) |
67 | 5, 8, 66 | cmpt 5232 |
. . 3
class (π β π« (π Γ π) β¦ β¦(π mPwSer π ) / πβ¦(π sSet β¨(leβndx), {β¨π₯, π¦β© β£ ({π₯, π¦} β (Baseβπ) β§ ([{β β (β0
βm π)
β£ (β‘β β β) β Fin} / π]βπ§ β π ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β π (π€(π <bag π)π§ β (π₯βπ€) = (π¦βπ€))) β¨ π₯ = π¦))}β©)) |
68 | 2, 3, 4, 4, 67 | cmpo 7411 |
. 2
class (π β V, π β V β¦ (π β π« (π Γ π) β¦ β¦(π mPwSer π ) / πβ¦(π sSet β¨(leβndx), {β¨π₯, π¦β© β£ ({π₯, π¦} β (Baseβπ) β§ ([{β β (β0
βm π)
β£ (β‘β β β) β Fin} / π]βπ§ β π ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β π (π€(π <bag π)π§ β (π₯βπ€) = (π¦βπ€))) β¨ π₯ = π¦))}β©))) |
69 | 1, 68 | wceq 1542 |
1
wff ordPwSer =
(π β V, π β V β¦ (π β π« (π Γ π) β¦ β¦(π mPwSer π ) / πβ¦(π sSet β¨(leβndx), {β¨π₯, π¦β© β£ ({π₯, π¦} β (Baseβπ) β§ ([{β β (β0
βm π)
β£ (β‘β β β) β Fin} / π]βπ§ β π ((π₯βπ§)(ltβπ )(π¦βπ§) β§ βπ€ β π (π€(π <bag π)π§ β (π₯βπ€) = (π¦βπ€))) β¨ π₯ = π¦))}β©))) |