Step | Hyp | Ref
| Expression |
1 | | sticksstones19.1 |
. . 3
β’ (π β π β
β0) |
2 | | sticksstones19.2 |
. . 3
β’ (π β πΎ β
β0) |
3 | | sticksstones19.3 |
. . 3
β’ π΄ = {π β£ (π:(1...πΎ)βΆβ0 β§
Ξ£π β (1...πΎ)(πβπ) = π)} |
4 | | sticksstones19.4 |
. . 3
β’ π΅ = {β β£ (β:πβΆβ0 β§
Ξ£π β π (ββπ) = π)} |
5 | | sticksstones19.5 |
. . 3
β’ (π β π:(1...πΎ)β1-1-ontoβπ) |
6 | | sticksstones19.6 |
. . 3
β’ πΉ = (π β π΄ β¦ (π₯ β π β¦ (πβ(β‘πβπ₯)))) |
7 | 1, 2, 3, 4, 5, 6 | sticksstones18 40618 |
. 2
β’ (π β πΉ:π΄βΆπ΅) |
8 | | sticksstones19.7 |
. . 3
β’ πΊ = (π β π΅ β¦ (π¦ β (1...πΎ) β¦ (πβ(πβπ¦)))) |
9 | 1, 2, 3, 4, 5, 8 | sticksstones17 40617 |
. 2
β’ (π β πΊ:π΅βΆπ΄) |
10 | 8 | a1i 11 |
. . . . 5
β’ ((π β§ π β π΄) β πΊ = (π β π΅ β¦ (π¦ β (1...πΎ) β¦ (πβ(πβπ¦))))) |
11 | | simplr 768 |
. . . . . . 7
β’ ((((π β§ π β π΄) β§ π = (πΉβπ)) β§ π¦ β (1...πΎ)) β π = (πΉβπ)) |
12 | 11 | fveq1d 6845 |
. . . . . 6
β’ ((((π β§ π β π΄) β§ π = (πΉβπ)) β§ π¦ β (1...πΎ)) β (πβ(πβπ¦)) = ((πΉβπ)β(πβπ¦))) |
13 | 12 | mpteq2dva 5206 |
. . . . 5
β’ (((π β§ π β π΄) β§ π = (πΉβπ)) β (π¦ β (1...πΎ) β¦ (πβ(πβπ¦))) = (π¦ β (1...πΎ) β¦ ((πΉβπ)β(πβπ¦)))) |
14 | 7 | ffvelcdmda 7036 |
. . . . 5
β’ ((π β§ π β π΄) β (πΉβπ) β π΅) |
15 | | fzfid 13884 |
. . . . . 6
β’ ((π β§ π β π΄) β (1...πΎ) β Fin) |
16 | 15 | mptexd 7175 |
. . . . 5
β’ ((π β§ π β π΄) β (π¦ β (1...πΎ) β¦ ((πΉβπ)β(πβπ¦))) β V) |
17 | 10, 13, 14, 16 | fvmptd 6956 |
. . . 4
β’ ((π β§ π β π΄) β (πΊβ(πΉβπ)) = (π¦ β (1...πΎ) β¦ ((πΉβπ)β(πβπ¦)))) |
18 | 6 | a1i 11 |
. . . . . . . . 9
β’ ((π β§ π β π΄ β§ π¦ β (1...πΎ)) β πΉ = (π β π΄ β¦ (π₯ β π β¦ (πβ(β‘πβπ₯))))) |
19 | 18 | fveq1d 6845 |
. . . . . . . 8
β’ ((π β§ π β π΄ β§ π¦ β (1...πΎ)) β (πΉβπ) = ((π β π΄ β¦ (π₯ β π β¦ (πβ(β‘πβπ₯))))βπ)) |
20 | 19 | fveq1d 6845 |
. . . . . . 7
β’ ((π β§ π β π΄ β§ π¦ β (1...πΎ)) β ((πΉβπ)β(πβπ¦)) = (((π β π΄ β¦ (π₯ β π β¦ (πβ(β‘πβπ₯))))βπ)β(πβπ¦))) |
21 | 20 | 3expa 1119 |
. . . . . 6
β’ (((π β§ π β π΄) β§ π¦ β (1...πΎ)) β ((πΉβπ)β(πβπ¦)) = (((π β π΄ β¦ (π₯ β π β¦ (πβ(β‘πβπ₯))))βπ)β(πβπ¦))) |
22 | 21 | mpteq2dva 5206 |
. . . . 5
β’ ((π β§ π β π΄) β (π¦ β (1...πΎ) β¦ ((πΉβπ)β(πβπ¦))) = (π¦ β (1...πΎ) β¦ (((π β π΄ β¦ (π₯ β π β¦ (πβ(β‘πβπ₯))))βπ)β(πβπ¦)))) |
23 | | eqidd 2734 |
. . . . . . . . 9
β’ (((π β§ π β π΄) β§ π¦ β (1...πΎ)) β (π β π΄ β¦ (π₯ β π β¦ (πβ(β‘πβπ₯)))) = (π β π΄ β¦ (π₯ β π β¦ (πβ(β‘πβπ₯))))) |
24 | | simplr 768 |
. . . . . . . . . . 11
β’
(((((π β§ π β π΄) β§ π¦ β (1...πΎ)) β§ π = π) β§ π₯ β π) β π = π) |
25 | 24 | fveq1d 6845 |
. . . . . . . . . 10
β’
(((((π β§ π β π΄) β§ π¦ β (1...πΎ)) β§ π = π) β§ π₯ β π) β (πβ(β‘πβπ₯)) = (πβ(β‘πβπ₯))) |
26 | 25 | mpteq2dva 5206 |
. . . . . . . . 9
β’ ((((π β§ π β π΄) β§ π¦ β (1...πΎ)) β§ π = π) β (π₯ β π β¦ (πβ(β‘πβπ₯))) = (π₯ β π β¦ (πβ(β‘πβπ₯)))) |
27 | | simplr 768 |
. . . . . . . . 9
β’ (((π β§ π β π΄) β§ π¦ β (1...πΎ)) β π β π΄) |
28 | | fzfid 13884 |
. . . . . . . . . . . . 13
β’ (π β (1...πΎ) β Fin) |
29 | | f1oenfi 9129 |
. . . . . . . . . . . . . . 15
β’
(((1...πΎ) β Fin
β§ π:(1...πΎ)β1-1-ontoβπ) β (1...πΎ) β π) |
30 | 28, 5, 29 | syl2anc 585 |
. . . . . . . . . . . . . 14
β’ (π β (1...πΎ) β π) |
31 | 30 | ensymd 8948 |
. . . . . . . . . . . . 13
β’ (π β π β (1...πΎ)) |
32 | | enfii 9136 |
. . . . . . . . . . . . 13
β’
(((1...πΎ) β Fin
β§ π β (1...πΎ)) β π β Fin) |
33 | 28, 31, 32 | syl2anc 585 |
. . . . . . . . . . . 12
β’ (π β π β Fin) |
34 | 33 | adantr 482 |
. . . . . . . . . . 11
β’ ((π β§ π β π΄) β π β Fin) |
35 | 34 | adantr 482 |
. . . . . . . . . 10
β’ (((π β§ π β π΄) β§ π¦ β (1...πΎ)) β π β Fin) |
36 | 35 | mptexd 7175 |
. . . . . . . . 9
β’ (((π β§ π β π΄) β§ π¦ β (1...πΎ)) β (π₯ β π β¦ (πβ(β‘πβπ₯))) β V) |
37 | 23, 26, 27, 36 | fvmptd 6956 |
. . . . . . . 8
β’ (((π β§ π β π΄) β§ π¦ β (1...πΎ)) β ((π β π΄ β¦ (π₯ β π β¦ (πβ(β‘πβπ₯))))βπ) = (π₯ β π β¦ (πβ(β‘πβπ₯)))) |
38 | 37 | fveq1d 6845 |
. . . . . . 7
β’ (((π β§ π β π΄) β§ π¦ β (1...πΎ)) β (((π β π΄ β¦ (π₯ β π β¦ (πβ(β‘πβπ₯))))βπ)β(πβπ¦)) = ((π₯ β π β¦ (πβ(β‘πβπ₯)))β(πβπ¦))) |
39 | 38 | mpteq2dva 5206 |
. . . . . 6
β’ ((π β§ π β π΄) β (π¦ β (1...πΎ) β¦ (((π β π΄ β¦ (π₯ β π β¦ (πβ(β‘πβπ₯))))βπ)β(πβπ¦))) = (π¦ β (1...πΎ) β¦ ((π₯ β π β¦ (πβ(β‘πβπ₯)))β(πβπ¦)))) |
40 | | eqidd 2734 |
. . . . . . . . 9
β’ (((π β§ π β π΄) β§ π¦ β (1...πΎ)) β (π₯ β π β¦ (πβ(β‘πβπ₯))) = (π₯ β π β¦ (πβ(β‘πβπ₯)))) |
41 | | simpr 486 |
. . . . . . . . . . 11
β’ ((((π β§ π β π΄) β§ π¦ β (1...πΎ)) β§ π₯ = (πβπ¦)) β π₯ = (πβπ¦)) |
42 | 41 | fveq2d 6847 |
. . . . . . . . . 10
β’ ((((π β§ π β π΄) β§ π¦ β (1...πΎ)) β§ π₯ = (πβπ¦)) β (β‘πβπ₯) = (β‘πβ(πβπ¦))) |
43 | 42 | fveq2d 6847 |
. . . . . . . . 9
β’ ((((π β§ π β π΄) β§ π¦ β (1...πΎ)) β§ π₯ = (πβπ¦)) β (πβ(β‘πβπ₯)) = (πβ(β‘πβ(πβπ¦)))) |
44 | | f1of 6785 |
. . . . . . . . . . . 12
β’ (π:(1...πΎ)β1-1-ontoβπ β π:(1...πΎ)βΆπ) |
45 | 5, 44 | syl 17 |
. . . . . . . . . . 11
β’ (π β π:(1...πΎ)βΆπ) |
46 | 45 | adantr 482 |
. . . . . . . . . 10
β’ ((π β§ π β π΄) β π:(1...πΎ)βΆπ) |
47 | 46 | ffvelcdmda 7036 |
. . . . . . . . 9
β’ (((π β§ π β π΄) β§ π¦ β (1...πΎ)) β (πβπ¦) β π) |
48 | | fvexd 6858 |
. . . . . . . . 9
β’ (((π β§ π β π΄) β§ π¦ β (1...πΎ)) β (πβ(β‘πβ(πβπ¦))) β V) |
49 | 40, 43, 47, 48 | fvmptd 6956 |
. . . . . . . 8
β’ (((π β§ π β π΄) β§ π¦ β (1...πΎ)) β ((π₯ β π β¦ (πβ(β‘πβπ₯)))β(πβπ¦)) = (πβ(β‘πβ(πβπ¦)))) |
50 | 49 | mpteq2dva 5206 |
. . . . . . 7
β’ ((π β§ π β π΄) β (π¦ β (1...πΎ) β¦ ((π₯ β π β¦ (πβ(β‘πβπ₯)))β(πβπ¦))) = (π¦ β (1...πΎ) β¦ (πβ(β‘πβ(πβπ¦))))) |
51 | 5 | ad2antrr 725 |
. . . . . . . . . . 11
β’ (((π β§ π β π΄) β§ π¦ β (1...πΎ)) β π:(1...πΎ)β1-1-ontoβπ) |
52 | | simpr 486 |
. . . . . . . . . . 11
β’ (((π β§ π β π΄) β§ π¦ β (1...πΎ)) β π¦ β (1...πΎ)) |
53 | | f1ocnvfv1 7223 |
. . . . . . . . . . 11
β’ ((π:(1...πΎ)β1-1-ontoβπ β§ π¦ β (1...πΎ)) β (β‘πβ(πβπ¦)) = π¦) |
54 | 51, 52, 53 | syl2anc 585 |
. . . . . . . . . 10
β’ (((π β§ π β π΄) β§ π¦ β (1...πΎ)) β (β‘πβ(πβπ¦)) = π¦) |
55 | 54 | fveq2d 6847 |
. . . . . . . . 9
β’ (((π β§ π β π΄) β§ π¦ β (1...πΎ)) β (πβ(β‘πβ(πβπ¦))) = (πβπ¦)) |
56 | 55 | mpteq2dva 5206 |
. . . . . . . 8
β’ ((π β§ π β π΄) β (π¦ β (1...πΎ) β¦ (πβ(β‘πβ(πβπ¦)))) = (π¦ β (1...πΎ) β¦ (πβπ¦))) |
57 | | simpr 486 |
. . . . . . . . . . . . . 14
β’ ((π β§ π β π΄) β π β π΄) |
58 | 3 | a1i 11 |
. . . . . . . . . . . . . 14
β’ ((π β§ π β π΄) β π΄ = {π β£ (π:(1...πΎ)βΆβ0 β§
Ξ£π β (1...πΎ)(πβπ) = π)}) |
59 | 57, 58 | eleqtrd 2836 |
. . . . . . . . . . . . 13
β’ ((π β§ π β π΄) β π β {π β£ (π:(1...πΎ)βΆβ0 β§
Ξ£π β (1...πΎ)(πβπ) = π)}) |
60 | | vex 3448 |
. . . . . . . . . . . . . 14
β’ π β V |
61 | | feq1 6650 |
. . . . . . . . . . . . . . 15
β’ (π = π β (π:(1...πΎ)βΆβ0 β π:(1...πΎ)βΆβ0)) |
62 | | simpl 484 |
. . . . . . . . . . . . . . . . . 18
β’ ((π = π β§ π β (1...πΎ)) β π = π) |
63 | 62 | fveq1d 6845 |
. . . . . . . . . . . . . . . . 17
β’ ((π = π β§ π β (1...πΎ)) β (πβπ) = (πβπ)) |
64 | 63 | sumeq2dv 15593 |
. . . . . . . . . . . . . . . 16
β’ (π = π β Ξ£π β (1...πΎ)(πβπ) = Ξ£π β (1...πΎ)(πβπ)) |
65 | 64 | eqeq1d 2735 |
. . . . . . . . . . . . . . 15
β’ (π = π β (Ξ£π β (1...πΎ)(πβπ) = π β Ξ£π β (1...πΎ)(πβπ) = π)) |
66 | 61, 65 | anbi12d 632 |
. . . . . . . . . . . . . 14
β’ (π = π β ((π:(1...πΎ)βΆβ0 β§
Ξ£π β (1...πΎ)(πβπ) = π) β (π:(1...πΎ)βΆβ0 β§
Ξ£π β (1...πΎ)(πβπ) = π))) |
67 | 60, 66 | elab 3631 |
. . . . . . . . . . . . 13
β’ (π β {π β£ (π:(1...πΎ)βΆβ0 β§
Ξ£π β (1...πΎ)(πβπ) = π)} β (π:(1...πΎ)βΆβ0 β§
Ξ£π β (1...πΎ)(πβπ) = π)) |
68 | 59, 67 | sylib 217 |
. . . . . . . . . . . 12
β’ ((π β§ π β π΄) β (π:(1...πΎ)βΆβ0 β§
Ξ£π β (1...πΎ)(πβπ) = π)) |
69 | 68 | simpld 496 |
. . . . . . . . . . 11
β’ ((π β§ π β π΄) β π:(1...πΎ)βΆβ0) |
70 | | ffn 6669 |
. . . . . . . . . . 11
β’ (π:(1...πΎ)βΆβ0 β π Fn (1...πΎ)) |
71 | 69, 70 | syl 17 |
. . . . . . . . . 10
β’ ((π β§ π β π΄) β π Fn (1...πΎ)) |
72 | | dffn5 6902 |
. . . . . . . . . 10
β’ (π Fn (1...πΎ) β π = (π¦ β (1...πΎ) β¦ (πβπ¦))) |
73 | 71, 72 | sylib 217 |
. . . . . . . . 9
β’ ((π β§ π β π΄) β π = (π¦ β (1...πΎ) β¦ (πβπ¦))) |
74 | 73 | eqcomd 2739 |
. . . . . . . 8
β’ ((π β§ π β π΄) β (π¦ β (1...πΎ) β¦ (πβπ¦)) = π) |
75 | 56, 74 | eqtrd 2773 |
. . . . . . 7
β’ ((π β§ π β π΄) β (π¦ β (1...πΎ) β¦ (πβ(β‘πβ(πβπ¦)))) = π) |
76 | 50, 75 | eqtrd 2773 |
. . . . . 6
β’ ((π β§ π β π΄) β (π¦ β (1...πΎ) β¦ ((π₯ β π β¦ (πβ(β‘πβπ₯)))β(πβπ¦))) = π) |
77 | 39, 76 | eqtrd 2773 |
. . . . 5
β’ ((π β§ π β π΄) β (π¦ β (1...πΎ) β¦ (((π β π΄ β¦ (π₯ β π β¦ (πβ(β‘πβπ₯))))βπ)β(πβπ¦))) = π) |
78 | 22, 77 | eqtrd 2773 |
. . . 4
β’ ((π β§ π β π΄) β (π¦ β (1...πΎ) β¦ ((πΉβπ)β(πβπ¦))) = π) |
79 | 17, 78 | eqtrd 2773 |
. . 3
β’ ((π β§ π β π΄) β (πΊβ(πΉβπ)) = π) |
80 | 79 | ralrimiva 3140 |
. 2
β’ (π β βπ β π΄ (πΊβ(πΉβπ)) = π) |
81 | 6 | a1i 11 |
. . . . 5
β’ ((π β§ π β π΅) β πΉ = (π β π΄ β¦ (π₯ β π β¦ (πβ(β‘πβπ₯))))) |
82 | | simplr 768 |
. . . . . . 7
β’ ((((π β§ π β π΅) β§ π = (πΊβπ)) β§ π₯ β π) β π = (πΊβπ)) |
83 | 82 | fveq1d 6845 |
. . . . . 6
β’ ((((π β§ π β π΅) β§ π = (πΊβπ)) β§ π₯ β π) β (πβ(β‘πβπ₯)) = ((πΊβπ)β(β‘πβπ₯))) |
84 | 83 | mpteq2dva 5206 |
. . . . 5
β’ (((π β§ π β π΅) β§ π = (πΊβπ)) β (π₯ β π β¦ (πβ(β‘πβπ₯))) = (π₯ β π β¦ ((πΊβπ)β(β‘πβπ₯)))) |
85 | 9 | ffvelcdmda 7036 |
. . . . 5
β’ ((π β§ π β π΅) β (πΊβπ) β π΄) |
86 | 33 | adantr 482 |
. . . . . 6
β’ ((π β§ π β π΅) β π β Fin) |
87 | 86 | mptexd 7175 |
. . . . 5
β’ ((π β§ π β π΅) β (π₯ β π β¦ ((πΊβπ)β(β‘πβπ₯))) β V) |
88 | 81, 84, 85, 87 | fvmptd 6956 |
. . . 4
β’ ((π β§ π β π΅) β (πΉβ(πΊβπ)) = (π₯ β π β¦ ((πΊβπ)β(β‘πβπ₯)))) |
89 | 8 | a1i 11 |
. . . . . . . 8
β’ (((π β§ π β π΅) β§ π₯ β π) β πΊ = (π β π΅ β¦ (π¦ β (1...πΎ) β¦ (πβ(πβπ¦))))) |
90 | | simplr 768 |
. . . . . . . . . 10
β’
(((((π β§ π β π΅) β§ π₯ β π) β§ π = π) β§ π¦ β (1...πΎ)) β π = π) |
91 | 90 | fveq1d 6845 |
. . . . . . . . 9
β’
(((((π β§ π β π΅) β§ π₯ β π) β§ π = π) β§ π¦ β (1...πΎ)) β (πβ(πβπ¦)) = (πβ(πβπ¦))) |
92 | 91 | mpteq2dva 5206 |
. . . . . . . 8
β’ ((((π β§ π β π΅) β§ π₯ β π) β§ π = π) β (π¦ β (1...πΎ) β¦ (πβ(πβπ¦))) = (π¦ β (1...πΎ) β¦ (πβ(πβπ¦)))) |
93 | | simplr 768 |
. . . . . . . 8
β’ (((π β§ π β π΅) β§ π₯ β π) β π β π΅) |
94 | | fzfid 13884 |
. . . . . . . . 9
β’ (((π β§ π β π΅) β§ π₯ β π) β (1...πΎ) β Fin) |
95 | 94 | mptexd 7175 |
. . . . . . . 8
β’ (((π β§ π β π΅) β§ π₯ β π) β (π¦ β (1...πΎ) β¦ (πβ(πβπ¦))) β V) |
96 | 89, 92, 93, 95 | fvmptd 6956 |
. . . . . . 7
β’ (((π β§ π β π΅) β§ π₯ β π) β (πΊβπ) = (π¦ β (1...πΎ) β¦ (πβ(πβπ¦)))) |
97 | 96 | fveq1d 6845 |
. . . . . 6
β’ (((π β§ π β π΅) β§ π₯ β π) β ((πΊβπ)β(β‘πβπ₯)) = ((π¦ β (1...πΎ) β¦ (πβ(πβπ¦)))β(β‘πβπ₯))) |
98 | 97 | mpteq2dva 5206 |
. . . . 5
β’ ((π β§ π β π΅) β (π₯ β π β¦ ((πΊβπ)β(β‘πβπ₯))) = (π₯ β π β¦ ((π¦ β (1...πΎ) β¦ (πβ(πβπ¦)))β(β‘πβπ₯)))) |
99 | | eqidd 2734 |
. . . . . . . 8
β’ (((π β§ π β π΅) β§ π₯ β π) β (π¦ β (1...πΎ) β¦ (πβ(πβπ¦))) = (π¦ β (1...πΎ) β¦ (πβ(πβπ¦)))) |
100 | | simpr 486 |
. . . . . . . . . 10
β’ ((((π β§ π β π΅) β§ π₯ β π) β§ π¦ = (β‘πβπ₯)) β π¦ = (β‘πβπ₯)) |
101 | 100 | fveq2d 6847 |
. . . . . . . . 9
β’ ((((π β§ π β π΅) β§ π₯ β π) β§ π¦ = (β‘πβπ₯)) β (πβπ¦) = (πβ(β‘πβπ₯))) |
102 | 101 | fveq2d 6847 |
. . . . . . . 8
β’ ((((π β§ π β π΅) β§ π₯ β π) β§ π¦ = (β‘πβπ₯)) β (πβ(πβπ¦)) = (πβ(πβ(β‘πβπ₯)))) |
103 | | f1ocnv 6797 |
. . . . . . . . . . . 12
β’ (π:(1...πΎ)β1-1-ontoβπ β β‘π:πβ1-1-ontoβ(1...πΎ)) |
104 | 5, 103 | syl 17 |
. . . . . . . . . . 11
β’ (π β β‘π:πβ1-1-ontoβ(1...πΎ)) |
105 | | f1of 6785 |
. . . . . . . . . . 11
β’ (β‘π:πβ1-1-ontoβ(1...πΎ) β β‘π:πβΆ(1...πΎ)) |
106 | 104, 105 | syl 17 |
. . . . . . . . . 10
β’ (π β β‘π:πβΆ(1...πΎ)) |
107 | 106 | adantr 482 |
. . . . . . . . 9
β’ ((π β§ π β π΅) β β‘π:πβΆ(1...πΎ)) |
108 | 107 | ffvelcdmda 7036 |
. . . . . . . 8
β’ (((π β§ π β π΅) β§ π₯ β π) β (β‘πβπ₯) β (1...πΎ)) |
109 | | fvexd 6858 |
. . . . . . . 8
β’ (((π β§ π β π΅) β§ π₯ β π) β (πβ(πβ(β‘πβπ₯))) β V) |
110 | 99, 102, 108, 109 | fvmptd 6956 |
. . . . . . 7
β’ (((π β§ π β π΅) β§ π₯ β π) β ((π¦ β (1...πΎ) β¦ (πβ(πβπ¦)))β(β‘πβπ₯)) = (πβ(πβ(β‘πβπ₯)))) |
111 | 110 | mpteq2dva 5206 |
. . . . . 6
β’ ((π β§ π β π΅) β (π₯ β π β¦ ((π¦ β (1...πΎ) β¦ (πβ(πβπ¦)))β(β‘πβπ₯))) = (π₯ β π β¦ (πβ(πβ(β‘πβπ₯))))) |
112 | 5 | ad2antrr 725 |
. . . . . . . . . 10
β’ (((π β§ π β π΅) β§ π₯ β π) β π:(1...πΎ)β1-1-ontoβπ) |
113 | | simpr 486 |
. . . . . . . . . 10
β’ (((π β§ π β π΅) β§ π₯ β π) β π₯ β π) |
114 | | f1ocnvfv2 7224 |
. . . . . . . . . 10
β’ ((π:(1...πΎ)β1-1-ontoβπ β§ π₯ β π) β (πβ(β‘πβπ₯)) = π₯) |
115 | 112, 113,
114 | syl2anc 585 |
. . . . . . . . 9
β’ (((π β§ π β π΅) β§ π₯ β π) β (πβ(β‘πβπ₯)) = π₯) |
116 | 115 | fveq2d 6847 |
. . . . . . . 8
β’ (((π β§ π β π΅) β§ π₯ β π) β (πβ(πβ(β‘πβπ₯))) = (πβπ₯)) |
117 | 116 | mpteq2dva 5206 |
. . . . . . 7
β’ ((π β§ π β π΅) β (π₯ β π β¦ (πβ(πβ(β‘πβπ₯)))) = (π₯ β π β¦ (πβπ₯))) |
118 | | simpr 486 |
. . . . . . . . . . . . 13
β’ ((π β§ π β π΅) β π β π΅) |
119 | 4 | a1i 11 |
. . . . . . . . . . . . 13
β’ ((π β§ π β π΅) β π΅ = {β β£ (β:πβΆβ0 β§
Ξ£π β π (ββπ) = π)}) |
120 | 118, 119 | eleqtrd 2836 |
. . . . . . . . . . . 12
β’ ((π β§ π β π΅) β π β {β β£ (β:πβΆβ0 β§
Ξ£π β π (ββπ) = π)}) |
121 | | vex 3448 |
. . . . . . . . . . . . 13
β’ π β V |
122 | | feq1 6650 |
. . . . . . . . . . . . . 14
β’ (β = π β (β:πβΆβ0 β π:πβΆβ0)) |
123 | | simpl 484 |
. . . . . . . . . . . . . . . . 17
β’ ((β = π β§ π β π) β β = π) |
124 | 123 | fveq1d 6845 |
. . . . . . . . . . . . . . . 16
β’ ((β = π β§ π β π) β (ββπ) = (πβπ)) |
125 | 124 | sumeq2dv 15593 |
. . . . . . . . . . . . . . 15
β’ (β = π β Ξ£π β π (ββπ) = Ξ£π β π (πβπ)) |
126 | 125 | eqeq1d 2735 |
. . . . . . . . . . . . . 14
β’ (β = π β (Ξ£π β π (ββπ) = π β Ξ£π β π (πβπ) = π)) |
127 | 122, 126 | anbi12d 632 |
. . . . . . . . . . . . 13
β’ (β = π β ((β:πβΆβ0 β§
Ξ£π β π (ββπ) = π) β (π:πβΆβ0 β§
Ξ£π β π (πβπ) = π))) |
128 | 121, 127 | elab 3631 |
. . . . . . . . . . . 12
β’ (π β {β β£ (β:πβΆβ0 β§
Ξ£π β π (ββπ) = π)} β (π:πβΆβ0 β§
Ξ£π β π (πβπ) = π)) |
129 | 120, 128 | sylib 217 |
. . . . . . . . . . 11
β’ ((π β§ π β π΅) β (π:πβΆβ0 β§
Ξ£π β π (πβπ) = π)) |
130 | 129 | simpld 496 |
. . . . . . . . . 10
β’ ((π β§ π β π΅) β π:πβΆβ0) |
131 | | ffn 6669 |
. . . . . . . . . 10
β’ (π:πβΆβ0 β π Fn π) |
132 | 130, 131 | syl 17 |
. . . . . . . . 9
β’ ((π β§ π β π΅) β π Fn π) |
133 | | dffn5 6902 |
. . . . . . . . 9
β’ (π Fn π β π = (π₯ β π β¦ (πβπ₯))) |
134 | 132, 133 | sylib 217 |
. . . . . . . 8
β’ ((π β§ π β π΅) β π = (π₯ β π β¦ (πβπ₯))) |
135 | 134 | eqcomd 2739 |
. . . . . . 7
β’ ((π β§ π β π΅) β (π₯ β π β¦ (πβπ₯)) = π) |
136 | 117, 135 | eqtrd 2773 |
. . . . . 6
β’ ((π β§ π β π΅) β (π₯ β π β¦ (πβ(πβ(β‘πβπ₯)))) = π) |
137 | 111, 136 | eqtrd 2773 |
. . . . 5
β’ ((π β§ π β π΅) β (π₯ β π β¦ ((π¦ β (1...πΎ) β¦ (πβ(πβπ¦)))β(β‘πβπ₯))) = π) |
138 | 98, 137 | eqtrd 2773 |
. . . 4
β’ ((π β§ π β π΅) β (π₯ β π β¦ ((πΊβπ)β(β‘πβπ₯))) = π) |
139 | 88, 138 | eqtrd 2773 |
. . 3
β’ ((π β§ π β π΅) β (πΉβ(πΊβπ)) = π) |
140 | 139 | ralrimiva 3140 |
. 2
β’ (π β βπ β π΅ (πΉβ(πΊβπ)) = π) |
141 | 7, 9, 80, 140 | 2fvidf1od 7245 |
1
β’ (π β πΉ:π΄β1-1-ontoβπ΅) |