Step | Hyp | Ref
| Expression |
1 | | ssun1 4137 |
. . . . . 6
β’ π΄ β (π΄ βͺ π΅) |
2 | | ssun2 4138 |
. . . . . . 7
β’ (π΄ βͺ π΅) β ({π} βͺ (π΄ βͺ π΅)) |
3 | | ordtval.1 |
. . . . . . . . . 10
β’ π = dom π
|
4 | | ordtval.2 |
. . . . . . . . . 10
β’ π΄ = ran (π₯ β π β¦ {π¦ β π β£ Β¬ π¦π
π₯}) |
5 | | ordtval.3 |
. . . . . . . . . 10
β’ π΅ = ran (π₯ β π β¦ {π¦ β π β£ Β¬ π₯π
π¦}) |
6 | 3, 4, 5 | ordtuni 22557 |
. . . . . . . . 9
β’ (π
β TosetRel β π = βͺ
({π} βͺ (π΄ βͺ π΅))) |
7 | | dmexg 7845 |
. . . . . . . . . 10
β’ (π
β TosetRel β dom
π
β
V) |
8 | 3, 7 | eqeltrid 2842 |
. . . . . . . . 9
β’ (π
β TosetRel β π β V) |
9 | 6, 8 | eqeltrrd 2839 |
. . . . . . . 8
β’ (π
β TosetRel β βͺ ({π}
βͺ (π΄ βͺ π΅)) β V) |
10 | | uniexb 7703 |
. . . . . . . 8
β’ (({π} βͺ (π΄ βͺ π΅)) β V β βͺ ({π}
βͺ (π΄ βͺ π΅)) β V) |
11 | 9, 10 | sylibr 233 |
. . . . . . 7
β’ (π
β TosetRel β ({π} βͺ (π΄ βͺ π΅)) β V) |
12 | | ssexg 5285 |
. . . . . . 7
β’ (((π΄ βͺ π΅) β ({π} βͺ (π΄ βͺ π΅)) β§ ({π} βͺ (π΄ βͺ π΅)) β V) β (π΄ βͺ π΅) β V) |
13 | 2, 11, 12 | sylancr 588 |
. . . . . 6
β’ (π
β TosetRel β (π΄ βͺ π΅) β V) |
14 | | ssexg 5285 |
. . . . . 6
β’ ((π΄ β (π΄ βͺ π΅) β§ (π΄ βͺ π΅) β V) β π΄ β V) |
15 | 1, 13, 14 | sylancr 588 |
. . . . 5
β’ (π
β TosetRel β π΄ β V) |
16 | | ssun2 4138 |
. . . . . 6
β’ π΅ β (π΄ βͺ π΅) |
17 | | ssexg 5285 |
. . . . . 6
β’ ((π΅ β (π΄ βͺ π΅) β§ (π΄ βͺ π΅) β V) β π΅ β V) |
18 | 16, 13, 17 | sylancr 588 |
. . . . 5
β’ (π
β TosetRel β π΅ β V) |
19 | | elfiun 9373 |
. . . . 5
β’ ((π΄ β V β§ π΅ β V) β (π§ β (fiβ(π΄ βͺ π΅)) β (π§ β (fiβπ΄) β¨ π§ β (fiβπ΅) β¨ βπ β (fiβπ΄)βπ β (fiβπ΅)π§ = (π β© π)))) |
20 | 15, 18, 19 | syl2anc 585 |
. . . 4
β’ (π
β TosetRel β (π§ β (fiβ(π΄ βͺ π΅)) β (π§ β (fiβπ΄) β¨ π§ β (fiβπ΅) β¨ βπ β (fiβπ΄)βπ β (fiβπ΅)π§ = (π β© π)))) |
21 | 3, 4 | ordtbaslem 22555 |
. . . . . . . 8
β’ (π
β TosetRel β
(fiβπ΄) = π΄) |
22 | 21, 1 | eqsstrdi 4003 |
. . . . . . 7
β’ (π
β TosetRel β
(fiβπ΄) β (π΄ βͺ π΅)) |
23 | | ssun1 4137 |
. . . . . . 7
β’ (π΄ βͺ π΅) β ((π΄ βͺ π΅) βͺ πΆ) |
24 | 22, 23 | sstrdi 3961 |
. . . . . 6
β’ (π
β TosetRel β
(fiβπ΄) β
((π΄ βͺ π΅) βͺ πΆ)) |
25 | 24 | sseld 3948 |
. . . . 5
β’ (π
β TosetRel β (π§ β (fiβπ΄) β π§ β ((π΄ βͺ π΅) βͺ πΆ))) |
26 | | cnvtsr 18484 |
. . . . . . . . . 10
β’ (π
β TosetRel β β‘π
β TosetRel ) |
27 | | df-rn 5649 |
. . . . . . . . . . 11
β’ ran π
= dom β‘π
|
28 | | eqid 2737 |
. . . . . . . . . . 11
β’ ran
(π₯ β ran π
β¦ {π¦ β ran π
β£ Β¬ π¦β‘π
π₯}) = ran (π₯ β ran π
β¦ {π¦ β ran π
β£ Β¬ π¦β‘π
π₯}) |
29 | 27, 28 | ordtbaslem 22555 |
. . . . . . . . . 10
β’ (β‘π
β TosetRel β (fiβran (π₯ β ran π
β¦ {π¦ β ran π
β£ Β¬ π¦β‘π
π₯})) = ran (π₯ β ran π
β¦ {π¦ β ran π
β£ Β¬ π¦β‘π
π₯})) |
30 | 26, 29 | syl 17 |
. . . . . . . . 9
β’ (π
β TosetRel β
(fiβran (π₯ β ran
π
β¦ {π¦ β ran π
β£ Β¬ π¦β‘π
π₯})) = ran (π₯ β ran π
β¦ {π¦ β ran π
β£ Β¬ π¦β‘π
π₯})) |
31 | | tsrps 18483 |
. . . . . . . . . . . . . 14
β’ (π
β TosetRel β π
β
PosetRel) |
32 | 3 | psrn 18471 |
. . . . . . . . . . . . . 14
β’ (π
β PosetRel β π = ran π
) |
33 | 31, 32 | syl 17 |
. . . . . . . . . . . . 13
β’ (π
β TosetRel β π = ran π
) |
34 | | vex 3452 |
. . . . . . . . . . . . . . . . . 18
β’ π¦ β V |
35 | | vex 3452 |
. . . . . . . . . . . . . . . . . 18
β’ π₯ β V |
36 | 34, 35 | brcnv 5843 |
. . . . . . . . . . . . . . . . 17
β’ (π¦β‘π
π₯ β π₯π
π¦) |
37 | 36 | bicomi 223 |
. . . . . . . . . . . . . . . 16
β’ (π₯π
π¦ β π¦β‘π
π₯) |
38 | 37 | notbii 320 |
. . . . . . . . . . . . . . 15
β’ (Β¬
π₯π
π¦ β Β¬ π¦β‘π
π₯) |
39 | 38 | a1i 11 |
. . . . . . . . . . . . . 14
β’ (π
β TosetRel β (Β¬
π₯π
π¦ β Β¬ π¦β‘π
π₯)) |
40 | 33, 39 | rabeqbidv 3427 |
. . . . . . . . . . . . 13
β’ (π
β TosetRel β {π¦ β π β£ Β¬ π₯π
π¦} = {π¦ β ran π
β£ Β¬ π¦β‘π
π₯}) |
41 | 33, 40 | mpteq12dv 5201 |
. . . . . . . . . . . 12
β’ (π
β TosetRel β (π₯ β π β¦ {π¦ β π β£ Β¬ π₯π
π¦}) = (π₯ β ran π
β¦ {π¦ β ran π
β£ Β¬ π¦β‘π
π₯})) |
42 | 41 | rneqd 5898 |
. . . . . . . . . . 11
β’ (π
β TosetRel β ran
(π₯ β π β¦ {π¦ β π β£ Β¬ π₯π
π¦}) = ran (π₯ β ran π
β¦ {π¦ β ran π
β£ Β¬ π¦β‘π
π₯})) |
43 | 5, 42 | eqtrid 2789 |
. . . . . . . . . 10
β’ (π
β TosetRel β π΅ = ran (π₯ β ran π
β¦ {π¦ β ran π
β£ Β¬ π¦β‘π
π₯})) |
44 | 43 | fveq2d 6851 |
. . . . . . . . 9
β’ (π
β TosetRel β
(fiβπ΅) =
(fiβran (π₯ β ran
π
β¦ {π¦ β ran π
β£ Β¬ π¦β‘π
π₯}))) |
45 | 30, 44, 43 | 3eqtr4d 2787 |
. . . . . . . 8
β’ (π
β TosetRel β
(fiβπ΅) = π΅) |
46 | 45, 16 | eqsstrdi 4003 |
. . . . . . 7
β’ (π
β TosetRel β
(fiβπ΅) β (π΄ βͺ π΅)) |
47 | 46, 23 | sstrdi 3961 |
. . . . . 6
β’ (π
β TosetRel β
(fiβπ΅) β
((π΄ βͺ π΅) βͺ πΆ)) |
48 | 47 | sseld 3948 |
. . . . 5
β’ (π
β TosetRel β (π§ β (fiβπ΅) β π§ β ((π΄ βͺ π΅) βͺ πΆ))) |
49 | | ssun2 4138 |
. . . . . . . 8
β’ πΆ β ((π΄ βͺ π΅) βͺ πΆ) |
50 | 21, 4 | eqtrdi 2793 |
. . . . . . . . . . . . . . 15
β’ (π
β TosetRel β
(fiβπ΄) = ran (π₯ β π β¦ {π¦ β π β£ Β¬ π¦π
π₯})) |
51 | 50 | eleq2d 2824 |
. . . . . . . . . . . . . 14
β’ (π
β TosetRel β (π β (fiβπ΄) β π β ran (π₯ β π β¦ {π¦ β π β£ Β¬ π¦π
π₯}))) |
52 | | breq2 5114 |
. . . . . . . . . . . . . . . . . . 19
β’ (π₯ = π β (π¦π
π₯ β π¦π
π)) |
53 | 52 | notbid 318 |
. . . . . . . . . . . . . . . . . 18
β’ (π₯ = π β (Β¬ π¦π
π₯ β Β¬ π¦π
π)) |
54 | 53 | rabbidv 3418 |
. . . . . . . . . . . . . . . . 17
β’ (π₯ = π β {π¦ β π β£ Β¬ π¦π
π₯} = {π¦ β π β£ Β¬ π¦π
π}) |
55 | 54 | cbvmptv 5223 |
. . . . . . . . . . . . . . . 16
β’ (π₯ β π β¦ {π¦ β π β£ Β¬ π¦π
π₯}) = (π β π β¦ {π¦ β π β£ Β¬ π¦π
π}) |
56 | 55 | elrnmpt 5916 |
. . . . . . . . . . . . . . 15
β’ (π β V β (π β ran (π₯ β π β¦ {π¦ β π β£ Β¬ π¦π
π₯}) β βπ β π π = {π¦ β π β£ Β¬ π¦π
π})) |
57 | 56 | elv 3454 |
. . . . . . . . . . . . . 14
β’ (π β ran (π₯ β π β¦ {π¦ β π β£ Β¬ π¦π
π₯}) β βπ β π π = {π¦ β π β£ Β¬ π¦π
π}) |
58 | 51, 57 | bitrdi 287 |
. . . . . . . . . . . . 13
β’ (π
β TosetRel β (π β (fiβπ΄) β βπ β π π = {π¦ β π β£ Β¬ π¦π
π})) |
59 | 45, 5 | eqtrdi 2793 |
. . . . . . . . . . . . . . 15
β’ (π
β TosetRel β
(fiβπ΅) = ran (π₯ β π β¦ {π¦ β π β£ Β¬ π₯π
π¦})) |
60 | 59 | eleq2d 2824 |
. . . . . . . . . . . . . 14
β’ (π
β TosetRel β (π β (fiβπ΅) β π β ran (π₯ β π β¦ {π¦ β π β£ Β¬ π₯π
π¦}))) |
61 | | breq1 5113 |
. . . . . . . . . . . . . . . . . . 19
β’ (π₯ = π β (π₯π
π¦ β ππ
π¦)) |
62 | 61 | notbid 318 |
. . . . . . . . . . . . . . . . . 18
β’ (π₯ = π β (Β¬ π₯π
π¦ β Β¬ ππ
π¦)) |
63 | 62 | rabbidv 3418 |
. . . . . . . . . . . . . . . . 17
β’ (π₯ = π β {π¦ β π β£ Β¬ π₯π
π¦} = {π¦ β π β£ Β¬ ππ
π¦}) |
64 | 63 | cbvmptv 5223 |
. . . . . . . . . . . . . . . 16
β’ (π₯ β π β¦ {π¦ β π β£ Β¬ π₯π
π¦}) = (π β π β¦ {π¦ β π β£ Β¬ ππ
π¦}) |
65 | 64 | elrnmpt 5916 |
. . . . . . . . . . . . . . 15
β’ (π β V β (π β ran (π₯ β π β¦ {π¦ β π β£ Β¬ π₯π
π¦}) β βπ β π π = {π¦ β π β£ Β¬ ππ
π¦})) |
66 | 65 | elv 3454 |
. . . . . . . . . . . . . 14
β’ (π β ran (π₯ β π β¦ {π¦ β π β£ Β¬ π₯π
π¦}) β βπ β π π = {π¦ β π β£ Β¬ ππ
π¦}) |
67 | 60, 66 | bitrdi 287 |
. . . . . . . . . . . . 13
β’ (π
β TosetRel β (π β (fiβπ΅) β βπ β π π = {π¦ β π β£ Β¬ ππ
π¦})) |
68 | 58, 67 | anbi12d 632 |
. . . . . . . . . . . 12
β’ (π
β TosetRel β ((π β (fiβπ΄) β§ π β (fiβπ΅)) β (βπ β π π = {π¦ β π β£ Β¬ π¦π
π} β§ βπ β π π = {π¦ β π β£ Β¬ ππ
π¦}))) |
69 | | reeanv 3220 |
. . . . . . . . . . . . 13
β’
(βπ β
π βπ β π (π = {π¦ β π β£ Β¬ π¦π
π} β§ π = {π¦ β π β£ Β¬ ππ
π¦}) β (βπ β π π = {π¦ β π β£ Β¬ π¦π
π} β§ βπ β π π = {π¦ β π β£ Β¬ ππ
π¦})) |
70 | | ineq12 4172 |
. . . . . . . . . . . . . . . 16
β’ ((π = {π¦ β π β£ Β¬ π¦π
π} β§ π = {π¦ β π β£ Β¬ ππ
π¦}) β (π β© π) = ({π¦ β π β£ Β¬ π¦π
π} β© {π¦ β π β£ Β¬ ππ
π¦})) |
71 | | inrab 4271 |
. . . . . . . . . . . . . . . 16
β’ ({π¦ β π β£ Β¬ π¦π
π} β© {π¦ β π β£ Β¬ ππ
π¦}) = {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)} |
72 | 70, 71 | eqtrdi 2793 |
. . . . . . . . . . . . . . 15
β’ ((π = {π¦ β π β£ Β¬ π¦π
π} β§ π = {π¦ β π β£ Β¬ ππ
π¦}) β (π β© π) = {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)}) |
73 | 72 | reximi 3088 |
. . . . . . . . . . . . . 14
β’
(βπ β
π (π = {π¦ β π β£ Β¬ π¦π
π} β§ π = {π¦ β π β£ Β¬ ππ
π¦}) β βπ β π (π β© π) = {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)}) |
74 | 73 | reximi 3088 |
. . . . . . . . . . . . 13
β’
(βπ β
π βπ β π (π = {π¦ β π β£ Β¬ π¦π
π} β§ π = {π¦ β π β£ Β¬ ππ
π¦}) β βπ β π βπ β π (π β© π) = {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)}) |
75 | 69, 74 | sylbir 234 |
. . . . . . . . . . . 12
β’
((βπ β
π π = {π¦ β π β£ Β¬ π¦π
π} β§ βπ β π π = {π¦ β π β£ Β¬ ππ
π¦}) β βπ β π βπ β π (π β© π) = {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)}) |
76 | 68, 75 | syl6bi 253 |
. . . . . . . . . . 11
β’ (π
β TosetRel β ((π β (fiβπ΄) β§ π β (fiβπ΅)) β βπ β π βπ β π (π β© π) = {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)})) |
77 | 76 | imp 408 |
. . . . . . . . . 10
β’ ((π
β TosetRel β§ (π β (fiβπ΄) β§ π β (fiβπ΅))) β βπ β π βπ β π (π β© π) = {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)}) |
78 | | vex 3452 |
. . . . . . . . . . . 12
β’ π β V |
79 | 78 | inex1 5279 |
. . . . . . . . . . 11
β’ (π β© π) β V |
80 | | eqid 2737 |
. . . . . . . . . . . 12
β’ (π β π, π β π β¦ {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)}) = (π β π, π β π β¦ {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)}) |
81 | 80 | elrnmpog 7496 |
. . . . . . . . . . 11
β’ ((π β© π) β V β ((π β© π) β ran (π β π, π β π β¦ {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)}) β βπ β π βπ β π (π β© π) = {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)})) |
82 | 79, 81 | ax-mp 5 |
. . . . . . . . . 10
β’ ((π β© π) β ran (π β π, π β π β¦ {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)}) β βπ β π βπ β π (π β© π) = {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)}) |
83 | 77, 82 | sylibr 233 |
. . . . . . . . 9
β’ ((π
β TosetRel β§ (π β (fiβπ΄) β§ π β (fiβπ΅))) β (π β© π) β ran (π β π, π β π β¦ {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)})) |
84 | | ordtval.4 |
. . . . . . . . 9
β’ πΆ = ran (π β π, π β π β¦ {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)}) |
85 | 83, 84 | eleqtrrdi 2849 |
. . . . . . . 8
β’ ((π
β TosetRel β§ (π β (fiβπ΄) β§ π β (fiβπ΅))) β (π β© π) β πΆ) |
86 | 49, 85 | sselid 3947 |
. . . . . . 7
β’ ((π
β TosetRel β§ (π β (fiβπ΄) β§ π β (fiβπ΅))) β (π β© π) β ((π΄ βͺ π΅) βͺ πΆ)) |
87 | | eleq1 2826 |
. . . . . . 7
β’ (π§ = (π β© π) β (π§ β ((π΄ βͺ π΅) βͺ πΆ) β (π β© π) β ((π΄ βͺ π΅) βͺ πΆ))) |
88 | 86, 87 | syl5ibrcom 247 |
. . . . . 6
β’ ((π
β TosetRel β§ (π β (fiβπ΄) β§ π β (fiβπ΅))) β (π§ = (π β© π) β π§ β ((π΄ βͺ π΅) βͺ πΆ))) |
89 | 88 | rexlimdvva 3206 |
. . . . 5
β’ (π
β TosetRel β
(βπ β
(fiβπ΄)βπ β (fiβπ΅)π§ = (π β© π) β π§ β ((π΄ βͺ π΅) βͺ πΆ))) |
90 | 25, 48, 89 | 3jaod 1429 |
. . . 4
β’ (π
β TosetRel β ((π§ β (fiβπ΄) β¨ π§ β (fiβπ΅) β¨ βπ β (fiβπ΄)βπ β (fiβπ΅)π§ = (π β© π)) β π§ β ((π΄ βͺ π΅) βͺ πΆ))) |
91 | 20, 90 | sylbid 239 |
. . 3
β’ (π
β TosetRel β (π§ β (fiβ(π΄ βͺ π΅)) β π§ β ((π΄ βͺ π΅) βͺ πΆ))) |
92 | 91 | ssrdv 3955 |
. 2
β’ (π
β TosetRel β
(fiβ(π΄ βͺ π΅)) β ((π΄ βͺ π΅) βͺ πΆ)) |
93 | | ssfii 9362 |
. . . 4
β’ ((π΄ βͺ π΅) β V β (π΄ βͺ π΅) β (fiβ(π΄ βͺ π΅))) |
94 | 13, 93 | syl 17 |
. . 3
β’ (π
β TosetRel β (π΄ βͺ π΅) β (fiβ(π΄ βͺ π΅))) |
95 | 94 | adantr 482 |
. . . . . . . . . 10
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β (π΄ βͺ π΅) β (fiβ(π΄ βͺ π΅))) |
96 | | simprl 770 |
. . . . . . . . . . . . . 14
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β π β π) |
97 | | eqidd 2738 |
. . . . . . . . . . . . . 14
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β {π¦ β π β£ Β¬ π¦π
π} = {π¦ β π β£ Β¬ π¦π
π}) |
98 | 54 | rspceeqv 3600 |
. . . . . . . . . . . . . 14
β’ ((π β π β§ {π¦ β π β£ Β¬ π¦π
π} = {π¦ β π β£ Β¬ π¦π
π}) β βπ₯ β π {π¦ β π β£ Β¬ π¦π
π} = {π¦ β π β£ Β¬ π¦π
π₯}) |
99 | 96, 97, 98 | syl2anc 585 |
. . . . . . . . . . . . 13
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β βπ₯ β π {π¦ β π β£ Β¬ π¦π
π} = {π¦ β π β£ Β¬ π¦π
π₯}) |
100 | 8 | adantr 482 |
. . . . . . . . . . . . . 14
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β π β V) |
101 | | rabexg 5293 |
. . . . . . . . . . . . . 14
β’ (π β V β {π¦ β π β£ Β¬ π¦π
π} β V) |
102 | | eqid 2737 |
. . . . . . . . . . . . . . 15
β’ (π₯ β π β¦ {π¦ β π β£ Β¬ π¦π
π₯}) = (π₯ β π β¦ {π¦ β π β£ Β¬ π¦π
π₯}) |
103 | 102 | elrnmpt 5916 |
. . . . . . . . . . . . . 14
β’ ({π¦ β π β£ Β¬ π¦π
π} β V β ({π¦ β π β£ Β¬ π¦π
π} β ran (π₯ β π β¦ {π¦ β π β£ Β¬ π¦π
π₯}) β βπ₯ β π {π¦ β π β£ Β¬ π¦π
π} = {π¦ β π β£ Β¬ π¦π
π₯})) |
104 | 100, 101,
103 | 3syl 18 |
. . . . . . . . . . . . 13
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β ({π¦ β π β£ Β¬ π¦π
π} β ran (π₯ β π β¦ {π¦ β π β£ Β¬ π¦π
π₯}) β βπ₯ β π {π¦ β π β£ Β¬ π¦π
π} = {π¦ β π β£ Β¬ π¦π
π₯})) |
105 | 99, 104 | mpbird 257 |
. . . . . . . . . . . 12
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β {π¦ β π β£ Β¬ π¦π
π} β ran (π₯ β π β¦ {π¦ β π β£ Β¬ π¦π
π₯})) |
106 | 105, 4 | eleqtrrdi 2849 |
. . . . . . . . . . 11
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β {π¦ β π β£ Β¬ π¦π
π} β π΄) |
107 | 1, 106 | sselid 3947 |
. . . . . . . . . 10
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β {π¦ β π β£ Β¬ π¦π
π} β (π΄ βͺ π΅)) |
108 | 95, 107 | sseldd 3950 |
. . . . . . . . 9
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β {π¦ β π β£ Β¬ π¦π
π} β (fiβ(π΄ βͺ π΅))) |
109 | | simprr 772 |
. . . . . . . . . . . . . 14
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β π β π) |
110 | | eqidd 2738 |
. . . . . . . . . . . . . 14
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β {π¦ β π β£ Β¬ ππ
π¦} = {π¦ β π β£ Β¬ ππ
π¦}) |
111 | 63 | rspceeqv 3600 |
. . . . . . . . . . . . . 14
β’ ((π β π β§ {π¦ β π β£ Β¬ ππ
π¦} = {π¦ β π β£ Β¬ ππ
π¦}) β βπ₯ β π {π¦ β π β£ Β¬ ππ
π¦} = {π¦ β π β£ Β¬ π₯π
π¦}) |
112 | 109, 110,
111 | syl2anc 585 |
. . . . . . . . . . . . 13
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β βπ₯ β π {π¦ β π β£ Β¬ ππ
π¦} = {π¦ β π β£ Β¬ π₯π
π¦}) |
113 | | rabexg 5293 |
. . . . . . . . . . . . . 14
β’ (π β V β {π¦ β π β£ Β¬ ππ
π¦} β V) |
114 | | eqid 2737 |
. . . . . . . . . . . . . . 15
β’ (π₯ β π β¦ {π¦ β π β£ Β¬ π₯π
π¦}) = (π₯ β π β¦ {π¦ β π β£ Β¬ π₯π
π¦}) |
115 | 114 | elrnmpt 5916 |
. . . . . . . . . . . . . 14
β’ ({π¦ β π β£ Β¬ ππ
π¦} β V β ({π¦ β π β£ Β¬ ππ
π¦} β ran (π₯ β π β¦ {π¦ β π β£ Β¬ π₯π
π¦}) β βπ₯ β π {π¦ β π β£ Β¬ ππ
π¦} = {π¦ β π β£ Β¬ π₯π
π¦})) |
116 | 100, 113,
115 | 3syl 18 |
. . . . . . . . . . . . 13
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β ({π¦ β π β£ Β¬ ππ
π¦} β ran (π₯ β π β¦ {π¦ β π β£ Β¬ π₯π
π¦}) β βπ₯ β π {π¦ β π β£ Β¬ ππ
π¦} = {π¦ β π β£ Β¬ π₯π
π¦})) |
117 | 112, 116 | mpbird 257 |
. . . . . . . . . . . 12
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β {π¦ β π β£ Β¬ ππ
π¦} β ran (π₯ β π β¦ {π¦ β π β£ Β¬ π₯π
π¦})) |
118 | 117, 5 | eleqtrrdi 2849 |
. . . . . . . . . . 11
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β {π¦ β π β£ Β¬ ππ
π¦} β π΅) |
119 | 16, 118 | sselid 3947 |
. . . . . . . . . 10
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β {π¦ β π β£ Β¬ ππ
π¦} β (π΄ βͺ π΅)) |
120 | 95, 119 | sseldd 3950 |
. . . . . . . . 9
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β {π¦ β π β£ Β¬ ππ
π¦} β (fiβ(π΄ βͺ π΅))) |
121 | | fiin 9365 |
. . . . . . . . 9
β’ (({π¦ β π β£ Β¬ π¦π
π} β (fiβ(π΄ βͺ π΅)) β§ {π¦ β π β£ Β¬ ππ
π¦} β (fiβ(π΄ βͺ π΅))) β ({π¦ β π β£ Β¬ π¦π
π} β© {π¦ β π β£ Β¬ ππ
π¦}) β (fiβ(π΄ βͺ π΅))) |
122 | 108, 120,
121 | syl2anc 585 |
. . . . . . . 8
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β ({π¦ β π β£ Β¬ π¦π
π} β© {π¦ β π β£ Β¬ ππ
π¦}) β (fiβ(π΄ βͺ π΅))) |
123 | 71, 122 | eqeltrrid 2843 |
. . . . . . 7
β’ ((π
β TosetRel β§ (π β π β§ π β π)) β {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)} β (fiβ(π΄ βͺ π΅))) |
124 | 123 | ralrimivva 3198 |
. . . . . 6
β’ (π
β TosetRel β
βπ β π βπ β π {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)} β (fiβ(π΄ βͺ π΅))) |
125 | 80 | fmpo 8005 |
. . . . . 6
β’
(βπ β
π βπ β π {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)} β (fiβ(π΄ βͺ π΅)) β (π β π, π β π β¦ {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)}):(π Γ π)βΆ(fiβ(π΄ βͺ π΅))) |
126 | 124, 125 | sylib 217 |
. . . . 5
β’ (π
β TosetRel β (π β π, π β π β¦ {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)}):(π Γ π)βΆ(fiβ(π΄ βͺ π΅))) |
127 | 126 | frnd 6681 |
. . . 4
β’ (π
β TosetRel β ran
(π β π, π β π β¦ {π¦ β π β£ (Β¬ π¦π
π β§ Β¬ ππ
π¦)}) β (fiβ(π΄ βͺ π΅))) |
128 | 84, 127 | eqsstrid 3997 |
. . 3
β’ (π
β TosetRel β πΆ β (fiβ(π΄ βͺ π΅))) |
129 | 94, 128 | unssd 4151 |
. 2
β’ (π
β TosetRel β ((π΄ βͺ π΅) βͺ πΆ) β (fiβ(π΄ βͺ π΅))) |
130 | 92, 129 | eqssd 3966 |
1
β’ (π
β TosetRel β
(fiβ(π΄ βͺ π΅)) = ((π΄ βͺ π΅) βͺ πΆ)) |