Step | Hyp | Ref
| Expression |
1 | | fveq2 6843 |
. . 3
β’ (π = β
β (rec(πΊ, β
)βπ) = (rec(πΊ, β
)ββ
)) |
2 | | fveq2 6843 |
. . 3
β’ (π = β
β
(π
1βπ) =
(π
1ββ
)) |
3 | | 2fveq3 6848 |
. . 3
β’ (π = β
β
(cardβ(π
1βπ)) =
(cardβ(π
1ββ
))) |
4 | 1, 2, 3 | f1oeq123d 6779 |
. 2
β’ (π = β
β ((rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ)) β (rec(πΊ,
β
)ββ
):(π
1ββ
)β1-1-ontoβ(cardβ(π
1ββ
)))) |
5 | | fveq2 6843 |
. . 3
β’ (π = π β (rec(πΊ, β
)βπ) = (rec(πΊ, β
)βπ)) |
6 | | fveq2 6843 |
. . 3
β’ (π = π β (π
1βπ) =
(π
1βπ)) |
7 | | 2fveq3 6848 |
. . 3
β’ (π = π β
(cardβ(π
1βπ)) =
(cardβ(π
1βπ))) |
8 | 5, 6, 7 | f1oeq123d 6779 |
. 2
β’ (π = π β ((rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ)) β (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ)))) |
9 | | fveq2 6843 |
. . 3
β’ (π = suc π β (rec(πΊ, β
)βπ) = (rec(πΊ, β
)βsuc π)) |
10 | | fveq2 6843 |
. . 3
β’ (π = suc π β (π
1βπ) =
(π
1βsuc π)) |
11 | | 2fveq3 6848 |
. . 3
β’ (π = suc π β
(cardβ(π
1βπ)) =
(cardβ(π
1βsuc π))) |
12 | 9, 10, 11 | f1oeq123d 6779 |
. 2
β’ (π = suc π β ((rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ)) β (rec(πΊ, β
)βsuc π):(π
1βsuc π)β1-1-ontoβ(cardβ(π
1βsuc
π)))) |
13 | | fveq2 6843 |
. . 3
β’ (π = π΄ β (rec(πΊ, β
)βπ) = (rec(πΊ, β
)βπ΄)) |
14 | | fveq2 6843 |
. . 3
β’ (π = π΄ β (π
1βπ) =
(π
1βπ΄)) |
15 | | 2fveq3 6848 |
. . 3
β’ (π = π΄ β
(cardβ(π
1βπ)) =
(cardβ(π
1βπ΄))) |
16 | 13, 14, 15 | f1oeq123d 6779 |
. 2
β’ (π = π΄ β ((rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ)) β (rec(πΊ, β
)βπ΄):(π
1βπ΄)β1-1-ontoβ(cardβ(π
1βπ΄)))) |
17 | | f1o0 6822 |
. . 3
β’
β
:β
β1-1-ontoββ
|
18 | | 0ex 5265 |
. . . . . 6
β’ β
β V |
19 | 18 | rdg0 8368 |
. . . . 5
β’
(rec(πΊ,
β
)ββ
) = β
|
20 | | f1oeq1 6773 |
. . . . 5
β’
((rec(πΊ,
β
)ββ
) = β
β ((rec(πΊ,
β
)ββ
):(π
1ββ
)β1-1-ontoβ(cardβ(π
1ββ
))
β β
:(π
1ββ
)β1-1-ontoβ(cardβ(π
1ββ
)))) |
21 | 19, 20 | ax-mp 5 |
. . . 4
β’
((rec(πΊ,
β
)ββ
):(π
1ββ
)β1-1-ontoβ(cardβ(π
1ββ
))
β β
:(π
1ββ
)β1-1-ontoβ(cardβ(π
1ββ
))) |
22 | | r10 9709 |
. . . . 5
β’
(π
1ββ
) = β
|
23 | 22 | fveq2i 6846 |
. . . . . 6
β’
(cardβ(π
1ββ
)) =
(cardββ
) |
24 | | card0 9899 |
. . . . . 6
β’
(cardββ
) = β
|
25 | 23, 24 | eqtri 2761 |
. . . . 5
β’
(cardβ(π
1ββ
)) =
β
|
26 | | f1oeq23 6776 |
. . . . 5
β’
(((π
1ββ
) = β
β§
(cardβ(π
1ββ
)) = β
) β
(β
:(π
1ββ
)β1-1-ontoβ(cardβ(π
1ββ
))
β β
:β
β1-1-ontoββ
)) |
27 | 22, 25, 26 | mp2an 691 |
. . . 4
β’
(β
:(π
1ββ
)β1-1-ontoβ(cardβ(π
1ββ
))
β β
:β
β1-1-ontoββ
) |
28 | 21, 27 | bitri 275 |
. . 3
β’
((rec(πΊ,
β
)ββ
):(π
1ββ
)β1-1-ontoβ(cardβ(π
1ββ
))
β β
:β
β1-1-ontoββ
) |
29 | 17, 28 | mpbir 230 |
. 2
β’
(rec(πΊ,
β
)ββ
):(π
1ββ
)β1-1-ontoβ(cardβ(π
1ββ
)) |
30 | | ackbij.f |
. . . . . . . . . 10
β’ πΉ = (π₯ β (π« Ο β© Fin) β¦
(cardββͺ π¦ β π₯ ({π¦} Γ π« π¦))) |
31 | 30 | ackbij1lem17 10177 |
. . . . . . . . 9
β’ πΉ:(π« Ο β©
Fin)β1-1βΟ |
32 | 31 | a1i 11 |
. . . . . . . 8
β’ (π β Ο β πΉ:(π« Ο β©
Fin)β1-1βΟ) |
33 | | r1fin 9714 |
. . . . . . . . . 10
β’ (π β Ο β
(π
1βπ) β Fin) |
34 | | ficardom 9902 |
. . . . . . . . . 10
β’
((π
1βπ) β Fin β
(cardβ(π
1βπ)) β Ο) |
35 | 33, 34 | syl 17 |
. . . . . . . . 9
β’ (π β Ο β
(cardβ(π
1βπ)) β Ο) |
36 | | ackbij2lem1 10160 |
. . . . . . . . 9
β’
((cardβ(π
1βπ)) β Ο β π«
(cardβ(π
1βπ)) β (π« Ο β©
Fin)) |
37 | 35, 36 | syl 17 |
. . . . . . . 8
β’ (π β Ο β π«
(cardβ(π
1βπ)) β (π« Ο β©
Fin)) |
38 | | f1ores 6799 |
. . . . . . . 8
β’ ((πΉ:(π« Ο β©
Fin)β1-1βΟ β§
π« (cardβ(π
1βπ)) β (π« Ο β© Fin))
β (πΉ βΎ π«
(cardβ(π
1βπ))):π«
(cardβ(π
1βπ))β1-1-ontoβ(πΉ β π«
(cardβ(π
1βπ)))) |
39 | 32, 37, 38 | syl2anc 585 |
. . . . . . 7
β’ (π β Ο β (πΉ βΎ π«
(cardβ(π
1βπ))):π«
(cardβ(π
1βπ))β1-1-ontoβ(πΉ β π«
(cardβ(π
1βπ)))) |
40 | 30 | ackbij1b 10180 |
. . . . . . . . . 10
β’
((cardβ(π
1βπ)) β Ο β (πΉ β π«
(cardβ(π
1βπ))) = (cardβπ«
(cardβ(π
1βπ)))) |
41 | 35, 40 | syl 17 |
. . . . . . . . 9
β’ (π β Ο β (πΉ β π«
(cardβ(π
1βπ))) = (cardβπ«
(cardβ(π
1βπ)))) |
42 | | ficardid 9903 |
. . . . . . . . . 10
β’
((π
1βπ) β Fin β
(cardβ(π
1βπ)) β (π
1βπ)) |
43 | | pwen 9097 |
. . . . . . . . . 10
β’
((cardβ(π
1βπ)) β (π
1βπ) β π«
(cardβ(π
1βπ)) β π«
(π
1βπ)) |
44 | | carden2b 9908 |
. . . . . . . . . 10
β’
(π« (cardβ(π
1βπ)) β π«
(π
1βπ) β (cardβπ«
(cardβ(π
1βπ))) = (cardβπ«
(π
1βπ))) |
45 | 33, 42, 43, 44 | 4syl 19 |
. . . . . . . . 9
β’ (π β Ο β
(cardβπ« (cardβ(π
1βπ))) = (cardβπ«
(π
1βπ))) |
46 | 41, 45 | eqtrd 2773 |
. . . . . . . 8
β’ (π β Ο β (πΉ β π«
(cardβ(π
1βπ))) = (cardβπ«
(π
1βπ))) |
47 | 46 | f1oeq3d 6782 |
. . . . . . 7
β’ (π β Ο β ((πΉ βΎ π«
(cardβ(π
1βπ))):π«
(cardβ(π
1βπ))β1-1-ontoβ(πΉ β π«
(cardβ(π
1βπ))) β (πΉ βΎ π«
(cardβ(π
1βπ))):π«
(cardβ(π
1βπ))β1-1-ontoβ(cardβπ«
(π
1βπ)))) |
48 | 39, 47 | mpbid 231 |
. . . . . 6
β’ (π β Ο β (πΉ βΎ π«
(cardβ(π
1βπ))):π«
(cardβ(π
1βπ))β1-1-ontoβ(cardβπ«
(π
1βπ))) |
49 | 48 | adantr 482 |
. . . . 5
β’ ((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β (πΉ βΎ π«
(cardβ(π
1βπ))):π«
(cardβ(π
1βπ))β1-1-ontoβ(cardβπ«
(π
1βπ))) |
50 | | f1opw 7610 |
. . . . . 6
β’
((rec(πΊ,
β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ)) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π)):π«
(π
1βπ)β1-1-ontoβπ«
(cardβ(π
1βπ))) |
51 | 50 | adantl 483 |
. . . . 5
β’ ((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π)):π«
(π
1βπ)β1-1-ontoβπ«
(cardβ(π
1βπ))) |
52 | | f1oco 6808 |
. . . . 5
β’ (((πΉ βΎ π«
(cardβ(π
1βπ))):π«
(cardβ(π
1βπ))β1-1-ontoβ(cardβπ«
(π
1βπ)) β§ (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π)):π«
(π
1βπ)β1-1-ontoβπ«
(cardβ(π
1βπ))) β ((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))):π«
(π
1βπ)β1-1-ontoβ(cardβπ«
(π
1βπ))) |
53 | 49, 51, 52 | syl2anc 585 |
. . . 4
β’ ((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β ((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))):π«
(π
1βπ)β1-1-ontoβ(cardβπ«
(π
1βπ))) |
54 | | frsuc 8384 |
. . . . . . . . 9
β’ (π β Ο β
((rec(πΊ, β
) βΎ
Ο)βsuc π) =
(πΊβ((rec(πΊ, β
) βΎ
Ο)βπ))) |
55 | | peano2 7828 |
. . . . . . . . . 10
β’ (π β Ο β suc π β
Ο) |
56 | 55 | fvresd 6863 |
. . . . . . . . 9
β’ (π β Ο β
((rec(πΊ, β
) βΎ
Ο)βsuc π) =
(rec(πΊ, β
)βsuc
π)) |
57 | | fvres 6862 |
. . . . . . . . . . 11
β’ (π β Ο β
((rec(πΊ, β
) βΎ
Ο)βπ) =
(rec(πΊ,
β
)βπ)) |
58 | 57 | fveq2d 6847 |
. . . . . . . . . 10
β’ (π β Ο β (πΊβ((rec(πΊ, β
) βΎ Ο)βπ)) = (πΊβ(rec(πΊ, β
)βπ))) |
59 | | fvex 6856 |
. . . . . . . . . . 11
β’
(rec(πΊ,
β
)βπ) β
V |
60 | | dmeq 5860 |
. . . . . . . . . . . . . 14
β’ (π₯ = (rec(πΊ, β
)βπ) β dom π₯ = dom (rec(πΊ, β
)βπ)) |
61 | 60 | pweqd 4578 |
. . . . . . . . . . . . 13
β’ (π₯ = (rec(πΊ, β
)βπ) β π« dom π₯ = π« dom (rec(πΊ, β
)βπ)) |
62 | | imaeq1 6009 |
. . . . . . . . . . . . . 14
β’ (π₯ = (rec(πΊ, β
)βπ) β (π₯ β π¦) = ((rec(πΊ, β
)βπ) β π¦)) |
63 | 62 | fveq2d 6847 |
. . . . . . . . . . . . 13
β’ (π₯ = (rec(πΊ, β
)βπ) β (πΉβ(π₯ β π¦)) = (πΉβ((rec(πΊ, β
)βπ) β π¦))) |
64 | 61, 63 | mpteq12dv 5197 |
. . . . . . . . . . . 12
β’ (π₯ = (rec(πΊ, β
)βπ) β (π¦ β π« dom π₯ β¦ (πΉβ(π₯ β π¦))) = (π¦ β π« dom (rec(πΊ, β
)βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦)))) |
65 | | ackbij.g |
. . . . . . . . . . . 12
β’ πΊ = (π₯ β V β¦ (π¦ β π« dom π₯ β¦ (πΉβ(π₯ β π¦)))) |
66 | 59 | dmex 7849 |
. . . . . . . . . . . . . 14
β’ dom
(rec(πΊ,
β
)βπ) β
V |
67 | 66 | pwex 5336 |
. . . . . . . . . . . . 13
β’ π«
dom (rec(πΊ,
β
)βπ) β
V |
68 | 67 | mptex 7174 |
. . . . . . . . . . . 12
β’ (π¦ β π« dom (rec(πΊ, β
)βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦))) β V |
69 | 64, 65, 68 | fvmpt 6949 |
. . . . . . . . . . 11
β’
((rec(πΊ,
β
)βπ) β V
β (πΊβ(rec(πΊ, β
)βπ)) = (π¦ β π« dom (rec(πΊ, β
)βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦)))) |
70 | 59, 69 | ax-mp 5 |
. . . . . . . . . 10
β’ (πΊβ(rec(πΊ, β
)βπ)) = (π¦ β π« dom (rec(πΊ, β
)βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦))) |
71 | 58, 70 | eqtrdi 2789 |
. . . . . . . . 9
β’ (π β Ο β (πΊβ((rec(πΊ, β
) βΎ Ο)βπ)) = (π¦ β π« dom (rec(πΊ, β
)βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦)))) |
72 | 54, 56, 71 | 3eqtr3d 2781 |
. . . . . . . 8
β’ (π β Ο β
(rec(πΊ, β
)βsuc
π) = (π¦ β π« dom (rec(πΊ, β
)βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦)))) |
73 | 72 | adantr 482 |
. . . . . . 7
β’ ((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β (rec(πΊ, β
)βsuc π) = (π¦ β π« dom (rec(πΊ, β
)βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦)))) |
74 | | f1odm 6789 |
. . . . . . . . . . 11
β’
((rec(πΊ,
β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ)) β dom (rec(πΊ, β
)βπ) =
(π
1βπ)) |
75 | 74 | adantl 483 |
. . . . . . . . . 10
β’ ((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β dom (rec(πΊ, β
)βπ) =
(π
1βπ)) |
76 | 75 | pweqd 4578 |
. . . . . . . . 9
β’ ((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β π« dom
(rec(πΊ,
β
)βπ) =
π« (π
1βπ)) |
77 | 76 | mpteq1d 5201 |
. . . . . . . 8
β’ ((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β (π¦ β π« dom (rec(πΊ, β
)βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦))) = (π¦ β π«
(π
1βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦)))) |
78 | | fvex 6856 |
. . . . . . . . . . 11
β’ (πΉβ((rec(πΊ, β
)βπ) β π¦)) β V |
79 | | eqid 2733 |
. . . . . . . . . . 11
β’ (π¦ β π«
(π
1βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦))) = (π¦ β π«
(π
1βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦))) |
80 | 78, 79 | fnmpti 6645 |
. . . . . . . . . 10
β’ (π¦ β π«
(π
1βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦))) Fn π«
(π
1βπ) |
81 | 80 | a1i 11 |
. . . . . . . . 9
β’ ((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β (π¦ β π«
(π
1βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦))) Fn π«
(π
1βπ)) |
82 | | f1ofn 6786 |
. . . . . . . . . 10
β’ (((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))):π«
(π
1βπ)β1-1-ontoβ(cardβπ«
(π
1βπ)) β ((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))) Fn π«
(π
1βπ)) |
83 | 53, 82 | syl 17 |
. . . . . . . . 9
β’ ((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β ((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))) Fn π«
(π
1βπ)) |
84 | | f1of 6785 |
. . . . . . . . . . . . . 14
β’ ((π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π)):π«
(π
1βπ)β1-1-ontoβπ«
(cardβ(π
1βπ)) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π)):π«
(π
1βπ)βΆπ«
(cardβ(π
1βπ))) |
85 | 51, 84 | syl 17 |
. . . . . . . . . . . . 13
β’ ((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π)):π«
(π
1βπ)βΆπ«
(cardβ(π
1βπ))) |
86 | 85 | ffvelcdmda 7036 |
. . . . . . . . . . . 12
β’ (((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β§ π β π«
(π
1βπ)) β ((π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))βπ) β π«
(cardβ(π
1βπ))) |
87 | 86 | fvresd 6863 |
. . . . . . . . . . 11
β’ (((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β§ π β π«
(π
1βπ)) β ((πΉ βΎ π«
(cardβ(π
1βπ)))β((π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))βπ)) = (πΉβ((π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))βπ))) |
88 | | imaeq2 6010 |
. . . . . . . . . . . . . 14
β’ (π = π β ((rec(πΊ, β
)βπ) β π) = ((rec(πΊ, β
)βπ) β π)) |
89 | | eqid 2733 |
. . . . . . . . . . . . . 14
β’ (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π)) = (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π)) |
90 | 59 | imaex 7854 |
. . . . . . . . . . . . . 14
β’
((rec(πΊ,
β
)βπ) β
π) β
V |
91 | 88, 89, 90 | fvmpt 6949 |
. . . . . . . . . . . . 13
β’ (π β π«
(π
1βπ) β ((π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))βπ) = ((rec(πΊ, β
)βπ) β π)) |
92 | 91 | adantl 483 |
. . . . . . . . . . . 12
β’ (((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β§ π β π«
(π
1βπ)) β ((π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))βπ) = ((rec(πΊ, β
)βπ) β π)) |
93 | 92 | fveq2d 6847 |
. . . . . . . . . . 11
β’ (((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β§ π β π«
(π
1βπ)) β (πΉβ((π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))βπ)) = (πΉβ((rec(πΊ, β
)βπ) β π))) |
94 | 87, 93 | eqtrd 2773 |
. . . . . . . . . 10
β’ (((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β§ π β π«
(π
1βπ)) β ((πΉ βΎ π«
(cardβ(π
1βπ)))β((π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))βπ)) = (πΉβ((rec(πΊ, β
)βπ) β π))) |
95 | | fvco3 6941 |
. . . . . . . . . . 11
β’ (((π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π)):π«
(π
1βπ)βΆπ«
(cardβ(π
1βπ)) β§ π β π«
(π
1βπ)) β (((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π)))βπ) = ((πΉ βΎ π«
(cardβ(π
1βπ)))β((π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))βπ))) |
96 | 85, 95 | sylan 581 |
. . . . . . . . . 10
β’ (((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β§ π β π«
(π
1βπ)) β (((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π)))βπ) = ((πΉ βΎ π«
(cardβ(π
1βπ)))β((π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))βπ))) |
97 | | imaeq2 6010 |
. . . . . . . . . . . . 13
β’ (π¦ = π β ((rec(πΊ, β
)βπ) β π¦) = ((rec(πΊ, β
)βπ) β π)) |
98 | 97 | fveq2d 6847 |
. . . . . . . . . . . 12
β’ (π¦ = π β (πΉβ((rec(πΊ, β
)βπ) β π¦)) = (πΉβ((rec(πΊ, β
)βπ) β π))) |
99 | | fvex 6856 |
. . . . . . . . . . . 12
β’ (πΉβ((rec(πΊ, β
)βπ) β π)) β V |
100 | 98, 79, 99 | fvmpt 6949 |
. . . . . . . . . . 11
β’ (π β π«
(π
1βπ) β ((π¦ β π«
(π
1βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦)))βπ) = (πΉβ((rec(πΊ, β
)βπ) β π))) |
101 | 100 | adantl 483 |
. . . . . . . . . 10
β’ (((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β§ π β π«
(π
1βπ)) β ((π¦ β π«
(π
1βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦)))βπ) = (πΉβ((rec(πΊ, β
)βπ) β π))) |
102 | 94, 96, 101 | 3eqtr4rd 2784 |
. . . . . . . . 9
β’ (((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β§ π β π«
(π
1βπ)) β ((π¦ β π«
(π
1βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦)))βπ) = (((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π)))βπ)) |
103 | 81, 83, 102 | eqfnfvd 6986 |
. . . . . . . 8
β’ ((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β (π¦ β π«
(π
1βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦))) = ((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π)))) |
104 | 77, 103 | eqtrd 2773 |
. . . . . . 7
β’ ((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β (π¦ β π« dom (rec(πΊ, β
)βπ) β¦ (πΉβ((rec(πΊ, β
)βπ) β π¦))) = ((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π)))) |
105 | 73, 104 | eqtrd 2773 |
. . . . . 6
β’ ((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β (rec(πΊ, β
)βsuc π) = ((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π)))) |
106 | | f1oeq1 6773 |
. . . . . 6
β’
((rec(πΊ,
β
)βsuc π) =
((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))) β ((rec(πΊ, β
)βsuc π):(π
1βsuc π)β1-1-ontoβ(cardβ(π
1βsuc
π)) β ((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))):(π
1βsuc π)β1-1-ontoβ(cardβ(π
1βsuc
π)))) |
107 | 105, 106 | syl 17 |
. . . . 5
β’ ((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β ((rec(πΊ, β
)βsuc π):(π
1βsuc π)β1-1-ontoβ(cardβ(π
1βsuc
π)) β ((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))):(π
1βsuc π)β1-1-ontoβ(cardβ(π
1βsuc
π)))) |
108 | | nnon 7809 |
. . . . . . . 8
β’ (π β Ο β π β On) |
109 | | r1suc 9711 |
. . . . . . . 8
β’ (π β On β
(π
1βsuc π) = π«
(π
1βπ)) |
110 | 108, 109 | syl 17 |
. . . . . . 7
β’ (π β Ο β
(π
1βsuc π) = π«
(π
1βπ)) |
111 | 110 | fveq2d 6847 |
. . . . . . 7
β’ (π β Ο β
(cardβ(π
1βsuc π)) = (cardβπ«
(π
1βπ))) |
112 | | f1oeq23 6776 |
. . . . . . 7
β’
(((π
1βsuc π) = π«
(π
1βπ) β§
(cardβ(π
1βsuc π)) = (cardβπ«
(π
1βπ))) β (((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))):(π
1βsuc π)β1-1-ontoβ(cardβ(π
1βsuc
π)) β ((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))):π«
(π
1βπ)β1-1-ontoβ(cardβπ«
(π
1βπ)))) |
113 | 110, 111,
112 | syl2anc 585 |
. . . . . 6
β’ (π β Ο β (((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))):(π
1βsuc π)β1-1-ontoβ(cardβ(π
1βsuc
π)) β ((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))):π«
(π
1βπ)β1-1-ontoβ(cardβπ«
(π
1βπ)))) |
114 | 113 | adantr 482 |
. . . . 5
β’ ((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β (((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))):(π
1βsuc π)β1-1-ontoβ(cardβ(π
1βsuc
π)) β ((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))):π«
(π
1βπ)β1-1-ontoβ(cardβπ«
(π
1βπ)))) |
115 | 107, 114 | bitrd 279 |
. . . 4
β’ ((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β ((rec(πΊ, β
)βsuc π):(π
1βsuc π)β1-1-ontoβ(cardβ(π
1βsuc
π)) β ((πΉ βΎ π«
(cardβ(π
1βπ))) β (π β π«
(π
1βπ) β¦ ((rec(πΊ, β
)βπ) β π))):π«
(π
1βπ)β1-1-ontoβ(cardβπ«
(π
1βπ)))) |
116 | 53, 115 | mpbird 257 |
. . 3
β’ ((π β Ο β§ (rec(πΊ, β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ))) β (rec(πΊ, β
)βsuc π):(π
1βsuc π)β1-1-ontoβ(cardβ(π
1βsuc
π))) |
117 | 116 | ex 414 |
. 2
β’ (π β Ο β
((rec(πΊ,
β
)βπ):(π
1βπ)β1-1-ontoβ(cardβ(π
1βπ)) β (rec(πΊ, β
)βsuc π):(π
1βsuc π)β1-1-ontoβ(cardβ(π
1βsuc
π)))) |
118 | 4, 8, 12, 16, 29, 117 | finds 7836 |
1
β’ (π΄ β Ο β
(rec(πΊ,
β
)βπ΄):(π
1βπ΄)β1-1-ontoβ(cardβ(π
1βπ΄))) |