Step | Hyp | Ref
| Expression |
1 | | fsupdm.6 |
. . 3
β’ π· = {π₯ β β©
π β π dom (πΉβπ) β£ βπ¦ β β βπ β π ((πΉβπ)βπ₯) β€ π¦} |
2 | | fsupdm.2 |
. . . 4
β’
β²π₯π |
3 | | nfcv 2903 |
. . . . 5
β’
β²π₯β |
4 | | nfcv 2903 |
. . . . . 6
β’
β²π₯π |
5 | | fsupdm.7 |
. . . . . . . . 9
β’ π» = (π β π β¦ (π β β β¦ {π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π})) |
6 | | nfrab1 3451 |
. . . . . . . . . . 11
β’
β²π₯{π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π} |
7 | 3, 6 | nfmpt 5255 |
. . . . . . . . . 10
β’
β²π₯(π β β β¦ {π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π}) |
8 | 4, 7 | nfmpt 5255 |
. . . . . . . . 9
β’
β²π₯(π β π β¦ (π β β β¦ {π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π})) |
9 | 5, 8 | nfcxfr 2901 |
. . . . . . . 8
β’
β²π₯π» |
10 | | nfcv 2903 |
. . . . . . . 8
β’
β²π₯π |
11 | 9, 10 | nffv 6901 |
. . . . . . 7
β’
β²π₯(π»βπ) |
12 | | nfcv 2903 |
. . . . . . 7
β’
β²π₯π |
13 | 11, 12 | nffv 6901 |
. . . . . 6
β’
β²π₯((π»βπ)βπ) |
14 | 4, 13 | nfiin 5028 |
. . . . 5
β’
β²π₯β© π β π ((π»βπ)βπ) |
15 | 3, 14 | nfiun 5027 |
. . . 4
β’
β²π₯βͺ π β β β© π β π ((π»βπ)βπ) |
16 | | fsupdm.3 |
. . . . . . . . . . 11
β’
β²ππ |
17 | | nfv 1917 |
. . . . . . . . . . 11
β’
β²π π₯ β β© π β π dom (πΉβπ) |
18 | 16, 17 | nfan 1902 |
. . . . . . . . . 10
β’
β²π(π β§ π₯ β β©
π β π dom (πΉβπ)) |
19 | | nfv 1917 |
. . . . . . . . . 10
β’
β²π π¦ β β |
20 | 18, 19 | nfan 1902 |
. . . . . . . . 9
β’
β²π((π β§ π₯ β β©
π β π dom (πΉβπ)) β§ π¦ β β) |
21 | | nfv 1917 |
. . . . . . . . 9
β’
β²πβπ β π ((πΉβπ)βπ₯) β€ π¦ |
22 | 20, 21 | nfan 1902 |
. . . . . . . 8
β’
β²π(((π β§ π₯ β β©
π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) |
23 | | fsupdm.1 |
. . . . . . . . . . . . 13
β’
β²ππ |
24 | | nfii1 5032 |
. . . . . . . . . . . . . 14
β’
β²πβ© π β π dom (πΉβπ) |
25 | 24 | nfcri 2890 |
. . . . . . . . . . . . 13
β’
β²π π₯ β β© π β π dom (πΉβπ) |
26 | 23, 25 | nfan 1902 |
. . . . . . . . . . . 12
β’
β²π(π β§ π₯ β β©
π β π dom (πΉβπ)) |
27 | | nfv 1917 |
. . . . . . . . . . . 12
β’
β²π π¦ β β |
28 | 26, 27 | nfan 1902 |
. . . . . . . . . . 11
β’
β²π((π β§ π₯ β β©
π β π dom (πΉβπ)) β§ π¦ β β) |
29 | | nfra1 3281 |
. . . . . . . . . . 11
β’
β²πβπ β π ((πΉβπ)βπ₯) β€ π¦ |
30 | 28, 29 | nfan 1902 |
. . . . . . . . . 10
β’
β²π(((π β§ π₯ β β©
π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) |
31 | | nfv 1917 |
. . . . . . . . . 10
β’
β²π π β β |
32 | | nfv 1917 |
. . . . . . . . . 10
β’
β²π π¦ < π |
33 | 30, 31, 32 | nf3an 1904 |
. . . . . . . . 9
β’
β²π((((π β§ π₯ β β©
π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) |
34 | | vex 3478 |
. . . . . . . . . 10
β’ π₯ β V |
35 | 34 | a1i 11 |
. . . . . . . . 9
β’
(((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β π₯ β V) |
36 | | simp-4r 782 |
. . . . . . . . . . . . 13
β’
(((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β π) β π₯ β β©
π β π dom (πΉβπ)) |
37 | 36 | 3ad2antl1 1185 |
. . . . . . . . . . . 12
β’
((((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β§ π β π) β π₯ β β©
π β π dom (πΉβπ)) |
38 | | simpr 485 |
. . . . . . . . . . . 12
β’
((((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β§ π β π) β π β π) |
39 | | eliinid 43790 |
. . . . . . . . . . . 12
β’ ((π₯ β β© π β π dom (πΉβπ) β§ π β π) β π₯ β dom (πΉβπ)) |
40 | 37, 38, 39 | syl2anc 584 |
. . . . . . . . . . 11
β’
((((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β§ π β π) β π₯ β dom (πΉβπ)) |
41 | | simp-4l 781 |
. . . . . . . . . . . . . . 15
β’
(((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β π) β π) |
42 | 41 | 3ad2antl1 1185 |
. . . . . . . . . . . . . 14
β’
((((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β§ π β π) β π) |
43 | | fsupdm.5 |
. . . . . . . . . . . . . 14
β’ ((π β§ π β π) β (πΉβπ):dom (πΉβπ)βΆβ*) |
44 | 42, 38, 43 | syl2anc 584 |
. . . . . . . . . . . . 13
β’
((((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β§ π β π) β (πΉβπ):dom (πΉβπ)βΆβ*) |
45 | 44, 40 | ffvelcdmd 7087 |
. . . . . . . . . . . 12
β’
((((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β§ π β π) β ((πΉβπ)βπ₯) β
β*) |
46 | | simpllr 774 |
. . . . . . . . . . . . . 14
β’
(((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β π) β π¦ β β) |
47 | 46 | rexrd 11263 |
. . . . . . . . . . . . 13
β’
(((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β π) β π¦ β β*) |
48 | 47 | 3ad2antl1 1185 |
. . . . . . . . . . . 12
β’
((((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β§ π β π) β π¦ β β*) |
49 | | simpl2 1192 |
. . . . . . . . . . . . 13
β’
((((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β§ π β π) β π β β) |
50 | 49 | nnxrd 43973 |
. . . . . . . . . . . 12
β’
((((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β§ π β π) β π β β*) |
51 | | simpl1r 1225 |
. . . . . . . . . . . . 13
β’
((((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β§ π β π) β βπ β π ((πΉβπ)βπ₯) β€ π¦) |
52 | | rspa 3245 |
. . . . . . . . . . . . 13
β’
((βπ β
π ((πΉβπ)βπ₯) β€ π¦ β§ π β π) β ((πΉβπ)βπ₯) β€ π¦) |
53 | 51, 38, 52 | syl2anc 584 |
. . . . . . . . . . . 12
β’
((((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β§ π β π) β ((πΉβπ)βπ₯) β€ π¦) |
54 | | simpl3 1193 |
. . . . . . . . . . . 12
β’
((((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β§ π β π) β π¦ < π) |
55 | 45, 48, 50, 53, 54 | xrlelttrd 13138 |
. . . . . . . . . . 11
β’
((((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β§ π β π) β ((πΉβπ)βπ₯) < π) |
56 | 40, 55 | rabidd 43839 |
. . . . . . . . . 10
β’
((((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β§ π β π) β π₯ β {π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π}) |
57 | | trud 1551 |
. . . . . . . . . . . . . 14
β’ (π β π β β€) |
58 | | id 22 |
. . . . . . . . . . . . . 14
β’ (π β π β π β π) |
59 | | nfcv 2903 |
. . . . . . . . . . . . . . 15
β’
β²ππ |
60 | | nnex 12217 |
. . . . . . . . . . . . . . . . 17
β’ β
β V |
61 | 60 | mptex 7224 |
. . . . . . . . . . . . . . . 16
β’ (π β β β¦ {π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π}) β V |
62 | 61 | a1i 11 |
. . . . . . . . . . . . . . 15
β’
((β€ β§ π
β π) β (π β β β¦ {π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π}) β V) |
63 | 59, 5, 62 | fvmpt2df 43967 |
. . . . . . . . . . . . . 14
β’
((β€ β§ π
β π) β (π»βπ) = (π β β β¦ {π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π})) |
64 | 57, 58, 63 | syl2anc 584 |
. . . . . . . . . . . . 13
β’ (π β π β (π»βπ) = (π β β β¦ {π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π})) |
65 | | fsupdm.4 |
. . . . . . . . . . . . . . . . 17
β’
β²π₯πΉ |
66 | 65, 10 | nffv 6901 |
. . . . . . . . . . . . . . . 16
β’
β²π₯(πΉβπ) |
67 | 66 | nfdm 5950 |
. . . . . . . . . . . . . . 15
β’
β²π₯dom
(πΉβπ) |
68 | | fvex 6904 |
. . . . . . . . . . . . . . . 16
β’ (πΉβπ) β V |
69 | 68 | dmex 7901 |
. . . . . . . . . . . . . . 15
β’ dom
(πΉβπ) β V |
70 | 67, 69 | rabexf 43813 |
. . . . . . . . . . . . . 14
β’ {π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π} β V |
71 | 70 | a1i 11 |
. . . . . . . . . . . . 13
β’ ((π β π β§ π β β) β {π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π} β V) |
72 | 64, 71 | fvmpt2d 7011 |
. . . . . . . . . . . 12
β’ ((π β π β§ π β β) β ((π»βπ)βπ) = {π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π}) |
73 | 72 | eqcomd 2738 |
. . . . . . . . . . 11
β’ ((π β π β§ π β β) β {π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π} = ((π»βπ)βπ)) |
74 | 38, 49, 73 | syl2anc 584 |
. . . . . . . . . 10
β’
((((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β§ π β π) β {π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π} = ((π»βπ)βπ)) |
75 | 56, 74 | eleqtrd 2835 |
. . . . . . . . 9
β’
((((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β§ π β π) β π₯ β ((π»βπ)βπ)) |
76 | 33, 35, 75 | eliind2 43809 |
. . . . . . . 8
β’
(((((π β§ π₯ β β© π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β§ π β β β§ π¦ < π) β π₯ β β©
π β π ((π»βπ)βπ)) |
77 | | arch 12468 |
. . . . . . . . 9
β’ (π¦ β β β
βπ β β
π¦ < π) |
78 | 77 | ad2antlr 725 |
. . . . . . . 8
β’ ((((π β§ π₯ β β©
π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β βπ β β π¦ < π) |
79 | 22, 76, 78 | reximdd 43831 |
. . . . . . 7
β’ ((((π β§ π₯ β β©
π β π dom (πΉβπ)) β§ π¦ β β) β§ βπ β π ((πΉβπ)βπ₯) β€ π¦) β βπ β β π₯ β β©
π β π ((π»βπ)βπ)) |
80 | 79 | rexlimdva2 3157 |
. . . . . 6
β’ ((π β§ π₯ β β©
π β π dom (πΉβπ)) β (βπ¦ β β βπ β π ((πΉβπ)βπ₯) β€ π¦ β βπ β β π₯ β β©
π β π ((π»βπ)βπ))) |
81 | 80 | 3impia 1117 |
. . . . 5
β’ ((π β§ π₯ β β©
π β π dom (πΉβπ) β§ βπ¦ β β βπ β π ((πΉβπ)βπ₯) β€ π¦) β βπ β β π₯ β β©
π β π ((π»βπ)βπ)) |
82 | | eliun 5001 |
. . . . 5
β’ (π₯ β βͺ π β β β© π β π ((π»βπ)βπ) β βπ β β π₯ β β©
π β π ((π»βπ)βπ)) |
83 | 81, 82 | sylibr 233 |
. . . 4
β’ ((π β§ π₯ β β©
π β π dom (πΉβπ) β§ βπ¦ β β βπ β π ((πΉβπ)βπ₯) β€ π¦) β π₯ β βͺ
π β β β© π β π ((π»βπ)βπ)) |
84 | 2, 15, 83 | rabssd 43821 |
. . 3
β’ (π β {π₯ β β©
π β π dom (πΉβπ) β£ βπ¦ β β βπ β π ((πΉβπ)βπ₯) β€ π¦} β βͺ
π β β β© π β π ((π»βπ)βπ)) |
85 | 1, 84 | eqsstrid 4030 |
. 2
β’ (π β π· β βͺ
π β β β© π β π ((π»βπ)βπ)) |
86 | | nfcv 2903 |
. . 3
β’
β²ππ· |
87 | | nfv 1917 |
. . . . 5
β’
β²π₯ π β β |
88 | 2, 87 | nfan 1902 |
. . . 4
β’
β²π₯(π β§ π β β) |
89 | | nfrab1 3451 |
. . . . 5
β’
β²π₯{π₯ β β©
π β π dom (πΉβπ) β£ βπ¦ β β βπ β π ((πΉβπ)βπ₯) β€ π¦} |
90 | 1, 89 | nfcxfr 2901 |
. . . 4
β’
β²π₯π· |
91 | 23, 31 | nfan 1902 |
. . . . . . . 8
β’
β²π(π β§ π β β) |
92 | | nfii1 5032 |
. . . . . . . . 9
β’
β²πβ© π β π ((π»βπ)βπ) |
93 | 92 | nfcri 2890 |
. . . . . . . 8
β’
β²π π₯ β β© π β π ((π»βπ)βπ) |
94 | 91, 93 | nfan 1902 |
. . . . . . 7
β’
β²π((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) |
95 | 34 | a1i 11 |
. . . . . . 7
β’ (((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β π₯ β V) |
96 | | eliinid 43790 |
. . . . . . . . . 10
β’ ((π₯ β β© π β π ((π»βπ)βπ) β§ π β π) β π₯ β ((π»βπ)βπ)) |
97 | 96 | adantll 712 |
. . . . . . . . 9
β’ ((((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β§ π β π) β π₯ β ((π»βπ)βπ)) |
98 | | simpr 485 |
. . . . . . . . . 10
β’ ((((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β§ π β π) β π β π) |
99 | | simpllr 774 |
. . . . . . . . . 10
β’ ((((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β§ π β π) β π β β) |
100 | 98, 99, 72 | syl2anc 584 |
. . . . . . . . 9
β’ ((((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β§ π β π) β ((π»βπ)βπ) = {π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π}) |
101 | 97, 100 | eleqtrd 2835 |
. . . . . . . 8
β’ ((((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β§ π β π) β π₯ β {π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π}) |
102 | | rabidim1 3453 |
. . . . . . . 8
β’ (π₯ β {π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π} β π₯ β dom (πΉβπ)) |
103 | 101, 102 | syl 17 |
. . . . . . 7
β’ ((((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β§ π β π) β π₯ β dom (πΉβπ)) |
104 | 94, 95, 103 | eliind2 43809 |
. . . . . 6
β’ (((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β π₯ β β©
π β π dom (πΉβπ)) |
105 | | nnre 12218 |
. . . . . . . 8
β’ (π β β β π β
β) |
106 | 105 | ad2antlr 725 |
. . . . . . 7
β’ (((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β π β β) |
107 | | breq2 5152 |
. . . . . . . . 9
β’ (π¦ = π β (((πΉβπ)βπ₯) β€ π¦ β ((πΉβπ)βπ₯) β€ π)) |
108 | 107 | ralbidv 3177 |
. . . . . . . 8
β’ (π¦ = π β (βπ β π ((πΉβπ)βπ₯) β€ π¦ β βπ β π ((πΉβπ)βπ₯) β€ π)) |
109 | 108 | adantl 482 |
. . . . . . 7
β’ ((((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β§ π¦ = π) β (βπ β π ((πΉβπ)βπ₯) β€ π¦ β βπ β π ((πΉβπ)βπ₯) β€ π)) |
110 | | simplll 773 |
. . . . . . . . . 10
β’ ((((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β§ π β π) β π) |
111 | 43 | 3adant3 1132 |
. . . . . . . . . . 11
β’ ((π β§ π β π β§ π₯ β dom (πΉβπ)) β (πΉβπ):dom (πΉβπ)βΆβ*) |
112 | | simp3 1138 |
. . . . . . . . . . 11
β’ ((π β§ π β π β§ π₯ β dom (πΉβπ)) β π₯ β dom (πΉβπ)) |
113 | 111, 112 | ffvelcdmd 7087 |
. . . . . . . . . 10
β’ ((π β§ π β π β§ π₯ β dom (πΉβπ)) β ((πΉβπ)βπ₯) β
β*) |
114 | 110, 98, 103, 113 | syl3anc 1371 |
. . . . . . . . 9
β’ ((((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β§ π β π) β ((πΉβπ)βπ₯) β
β*) |
115 | 99 | nnxrd 43973 |
. . . . . . . . 9
β’ ((((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β§ π β π) β π β β*) |
116 | | rabidim2 43781 |
. . . . . . . . . 10
β’ (π₯ β {π₯ β dom (πΉβπ) β£ ((πΉβπ)βπ₯) < π} β ((πΉβπ)βπ₯) < π) |
117 | 101, 116 | syl 17 |
. . . . . . . . 9
β’ ((((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β§ π β π) β ((πΉβπ)βπ₯) < π) |
118 | 114, 115,
117 | xrltled 13128 |
. . . . . . . 8
β’ ((((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β§ π β π) β ((πΉβπ)βπ₯) β€ π) |
119 | 94, 118 | ralrimia 3255 |
. . . . . . 7
β’ (((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β βπ β π ((πΉβπ)βπ₯) β€ π) |
120 | 106, 109,
119 | rspcedvd 3614 |
. . . . . 6
β’ (((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β βπ¦ β β βπ β π ((πΉβπ)βπ₯) β€ π¦) |
121 | 104, 120 | rabidd 43839 |
. . . . 5
β’ (((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β π₯ β {π₯ β β©
π β π dom (πΉβπ) β£ βπ¦ β β βπ β π ((πΉβπ)βπ₯) β€ π¦}) |
122 | 121, 1 | eleqtrrdi 2844 |
. . . 4
β’ (((π β§ π β β) β§ π₯ β β©
π β π ((π»βπ)βπ)) β π₯ β π·) |
123 | 88, 14, 90, 122 | ssdf2 43820 |
. . 3
β’ ((π β§ π β β) β β© π β π ((π»βπ)βπ) β π·) |
124 | 16, 86, 123 | iunssdf 43840 |
. 2
β’ (π β βͺ π β β β© π β π ((π»βπ)βπ) β π·) |
125 | 85, 124 | eqssd 3999 |
1
β’ (π β π· = βͺ π β β β© π β π ((π»βπ)βπ)) |