Step | Hyp | Ref
| Expression |
1 | | tailfb.1 |
. . . . 5
β’ π = dom π· |
2 | 1 | tailf 35308 |
. . . 4
β’ (π· β DirRel β
(tailβπ·):πβΆπ« π) |
3 | 2 | frnd 6726 |
. . 3
β’ (π· β DirRel β ran
(tailβπ·) β
π« π) |
4 | 3 | adantr 482 |
. 2
β’ ((π· β DirRel β§ π β β
) β ran
(tailβπ·) β
π« π) |
5 | | n0 4347 |
. . . . 5
β’ (π β β
β
βπ₯ π₯ β π) |
6 | | ffn 6718 |
. . . . . . . 8
β’
((tailβπ·):πβΆπ« π β (tailβπ·) Fn π) |
7 | | fnfvelrn 7083 |
. . . . . . . . 9
β’
(((tailβπ·) Fn
π β§ π₯ β π) β ((tailβπ·)βπ₯) β ran (tailβπ·)) |
8 | 7 | ex 414 |
. . . . . . . 8
β’
((tailβπ·) Fn
π β (π₯ β π β ((tailβπ·)βπ₯) β ran (tailβπ·))) |
9 | 2, 6, 8 | 3syl 18 |
. . . . . . 7
β’ (π· β DirRel β (π₯ β π β ((tailβπ·)βπ₯) β ran (tailβπ·))) |
10 | | ne0i 4335 |
. . . . . . 7
β’
(((tailβπ·)βπ₯) β ran (tailβπ·) β ran (tailβπ·) β β
) |
11 | 9, 10 | syl6 35 |
. . . . . 6
β’ (π· β DirRel β (π₯ β π β ran (tailβπ·) β β
)) |
12 | 11 | exlimdv 1937 |
. . . . 5
β’ (π· β DirRel β
(βπ₯ π₯ β π β ran (tailβπ·) β β
)) |
13 | 5, 12 | biimtrid 241 |
. . . 4
β’ (π· β DirRel β (π β β
β ran
(tailβπ·) β
β
)) |
14 | 13 | imp 408 |
. . 3
β’ ((π· β DirRel β§ π β β
) β ran
(tailβπ·) β
β
) |
15 | 1 | tailini 35309 |
. . . . . . . 8
β’ ((π· β DirRel β§ π₯ β π) β π₯ β ((tailβπ·)βπ₯)) |
16 | | n0i 4334 |
. . . . . . . 8
β’ (π₯ β ((tailβπ·)βπ₯) β Β¬ ((tailβπ·)βπ₯) = β
) |
17 | 15, 16 | syl 17 |
. . . . . . 7
β’ ((π· β DirRel β§ π₯ β π) β Β¬ ((tailβπ·)βπ₯) = β
) |
18 | 17 | nrexdv 3150 |
. . . . . 6
β’ (π· β DirRel β Β¬
βπ₯ β π ((tailβπ·)βπ₯) = β
) |
19 | 18 | adantr 482 |
. . . . 5
β’ ((π· β DirRel β§ π β β
) β Β¬
βπ₯ β π ((tailβπ·)βπ₯) = β
) |
20 | | fvelrnb 6953 |
. . . . . . 7
β’
((tailβπ·) Fn
π β (β
β
ran (tailβπ·) β
βπ₯ β π ((tailβπ·)βπ₯) = β
)) |
21 | 2, 6, 20 | 3syl 18 |
. . . . . 6
β’ (π· β DirRel β (β
β ran (tailβπ·)
β βπ₯ β
π ((tailβπ·)βπ₯) = β
)) |
22 | 21 | adantr 482 |
. . . . 5
β’ ((π· β DirRel β§ π β β
) β (β
β ran (tailβπ·)
β βπ₯ β
π ((tailβπ·)βπ₯) = β
)) |
23 | 19, 22 | mtbird 325 |
. . . 4
β’ ((π· β DirRel β§ π β β
) β Β¬
β
β ran (tailβπ·)) |
24 | | df-nel 3048 |
. . . 4
β’ (β
β ran (tailβπ·)
β Β¬ β
β ran (tailβπ·)) |
25 | 23, 24 | sylibr 233 |
. . 3
β’ ((π· β DirRel β§ π β β
) β β
β ran (tailβπ·)) |
26 | | fvelrnb 6953 |
. . . . . . . 8
β’
((tailβπ·) Fn
π β (π₯ β ran (tailβπ·) β βπ’ β π ((tailβπ·)βπ’) = π₯)) |
27 | | fvelrnb 6953 |
. . . . . . . 8
β’
((tailβπ·) Fn
π β (π¦ β ran (tailβπ·) β βπ£ β π ((tailβπ·)βπ£) = π¦)) |
28 | 26, 27 | anbi12d 632 |
. . . . . . 7
β’
((tailβπ·) Fn
π β ((π₯ β ran (tailβπ·) β§ π¦ β ran (tailβπ·)) β (βπ’ β π ((tailβπ·)βπ’) = π₯ β§ βπ£ β π ((tailβπ·)βπ£) = π¦))) |
29 | 2, 6, 28 | 3syl 18 |
. . . . . 6
β’ (π· β DirRel β ((π₯ β ran (tailβπ·) β§ π¦ β ran (tailβπ·)) β (βπ’ β π ((tailβπ·)βπ’) = π₯ β§ βπ£ β π ((tailβπ·)βπ£) = π¦))) |
30 | | reeanv 3227 |
. . . . . . 7
β’
(βπ’ β
π βπ£ β π (((tailβπ·)βπ’) = π₯ β§ ((tailβπ·)βπ£) = π¦) β (βπ’ β π ((tailβπ·)βπ’) = π₯ β§ βπ£ β π ((tailβπ·)βπ£) = π¦)) |
31 | 1 | dirge 18556 |
. . . . . . . . . . 11
β’ ((π· β DirRel β§ π’ β π β§ π£ β π) β βπ€ β π (π’π·π€ β§ π£π·π€)) |
32 | 31 | 3expb 1121 |
. . . . . . . . . 10
β’ ((π· β DirRel β§ (π’ β π β§ π£ β π)) β βπ€ β π (π’π·π€ β§ π£π·π€)) |
33 | 2, 6 | syl 17 |
. . . . . . . . . . . . 13
β’ (π· β DirRel β
(tailβπ·) Fn π) |
34 | | fnfvelrn 7083 |
. . . . . . . . . . . . 13
β’
(((tailβπ·) Fn
π β§ π€ β π) β ((tailβπ·)βπ€) β ran (tailβπ·)) |
35 | 33, 34 | sylan 581 |
. . . . . . . . . . . 12
β’ ((π· β DirRel β§ π€ β π) β ((tailβπ·)βπ€) β ran (tailβπ·)) |
36 | 35 | ad2ant2r 746 |
. . . . . . . . . . 11
β’ (((π· β DirRel β§ (π’ β π β§ π£ β π)) β§ (π€ β π β§ (π’π·π€ β§ π£π·π€))) β ((tailβπ·)βπ€) β ran (tailβπ·)) |
37 | | dirtr 18555 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (((π· β DirRel β§ π₯ β V) β§ (π’π·π€ β§ π€π·π₯)) β π’π·π₯) |
38 | 37 | exp32 422 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((π· β DirRel β§ π₯ β V) β (π’π·π€ β (π€π·π₯ β π’π·π₯))) |
39 | 38 | elvd 3482 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π· β DirRel β (π’π·π€ β (π€π·π₯ β π’π·π₯))) |
40 | 39 | com23 86 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π· β DirRel β (π€π·π₯ β (π’π·π€ β π’π·π₯))) |
41 | 40 | imp 408 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π· β DirRel β§ π€π·π₯) β (π’π·π€ β π’π·π₯)) |
42 | 41 | ad2ant2rl 748 |
. . . . . . . . . . . . . . . . . 18
β’ (((π· β DirRel β§ (π’ β π β§ π£ β π)) β§ (π€ β π β§ π€π·π₯)) β (π’π·π€ β π’π·π₯)) |
43 | | dirtr 18555 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (((π· β DirRel β§ π₯ β V) β§ (π£π·π€ β§ π€π·π₯)) β π£π·π₯) |
44 | 43 | exp32 422 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((π· β DirRel β§ π₯ β V) β (π£π·π€ β (π€π·π₯ β π£π·π₯))) |
45 | 44 | elvd 3482 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π· β DirRel β (π£π·π€ β (π€π·π₯ β π£π·π₯))) |
46 | 45 | com23 86 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π· β DirRel β (π€π·π₯ β (π£π·π€ β π£π·π₯))) |
47 | 46 | imp 408 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π· β DirRel β§ π€π·π₯) β (π£π·π€ β π£π·π₯)) |
48 | 47 | ad2ant2rl 748 |
. . . . . . . . . . . . . . . . . 18
β’ (((π· β DirRel β§ (π’ β π β§ π£ β π)) β§ (π€ β π β§ π€π·π₯)) β (π£π·π€ β π£π·π₯)) |
49 | 42, 48 | anim12d 610 |
. . . . . . . . . . . . . . . . 17
β’ (((π· β DirRel β§ (π’ β π β§ π£ β π)) β§ (π€ β π β§ π€π·π₯)) β ((π’π·π€ β§ π£π·π€) β (π’π·π₯ β§ π£π·π₯))) |
50 | 49 | expr 458 |
. . . . . . . . . . . . . . . 16
β’ (((π· β DirRel β§ (π’ β π β§ π£ β π)) β§ π€ β π) β (π€π·π₯ β ((π’π·π€ β§ π£π·π€) β (π’π·π₯ β§ π£π·π₯)))) |
51 | 50 | com23 86 |
. . . . . . . . . . . . . . 15
β’ (((π· β DirRel β§ (π’ β π β§ π£ β π)) β§ π€ β π) β ((π’π·π€ β§ π£π·π€) β (π€π·π₯ β (π’π·π₯ β§ π£π·π₯)))) |
52 | 51 | impr 456 |
. . . . . . . . . . . . . 14
β’ (((π· β DirRel β§ (π’ β π β§ π£ β π)) β§ (π€ β π β§ (π’π·π€ β§ π£π·π€))) β (π€π·π₯ β (π’π·π₯ β§ π£π·π₯))) |
53 | | vex 3479 |
. . . . . . . . . . . . . . . 16
β’ π₯ β V |
54 | 1 | eltail 35307 |
. . . . . . . . . . . . . . . 16
β’ ((π· β DirRel β§ π€ β π β§ π₯ β V) β (π₯ β ((tailβπ·)βπ€) β π€π·π₯)) |
55 | 53, 54 | mp3an3 1451 |
. . . . . . . . . . . . . . 15
β’ ((π· β DirRel β§ π€ β π) β (π₯ β ((tailβπ·)βπ€) β π€π·π₯)) |
56 | 55 | ad2ant2r 746 |
. . . . . . . . . . . . . 14
β’ (((π· β DirRel β§ (π’ β π β§ π£ β π)) β§ (π€ β π β§ (π’π·π€ β§ π£π·π€))) β (π₯ β ((tailβπ·)βπ€) β π€π·π₯)) |
57 | 1 | eltail 35307 |
. . . . . . . . . . . . . . . . . 18
β’ ((π· β DirRel β§ π’ β π β§ π₯ β V) β (π₯ β ((tailβπ·)βπ’) β π’π·π₯)) |
58 | 53, 57 | mp3an3 1451 |
. . . . . . . . . . . . . . . . 17
β’ ((π· β DirRel β§ π’ β π) β (π₯ β ((tailβπ·)βπ’) β π’π·π₯)) |
59 | 58 | adantrr 716 |
. . . . . . . . . . . . . . . 16
β’ ((π· β DirRel β§ (π’ β π β§ π£ β π)) β (π₯ β ((tailβπ·)βπ’) β π’π·π₯)) |
60 | 1 | eltail 35307 |
. . . . . . . . . . . . . . . . . 18
β’ ((π· β DirRel β§ π£ β π β§ π₯ β V) β (π₯ β ((tailβπ·)βπ£) β π£π·π₯)) |
61 | 53, 60 | mp3an3 1451 |
. . . . . . . . . . . . . . . . 17
β’ ((π· β DirRel β§ π£ β π) β (π₯ β ((tailβπ·)βπ£) β π£π·π₯)) |
62 | 61 | adantrl 715 |
. . . . . . . . . . . . . . . 16
β’ ((π· β DirRel β§ (π’ β π β§ π£ β π)) β (π₯ β ((tailβπ·)βπ£) β π£π·π₯)) |
63 | 59, 62 | anbi12d 632 |
. . . . . . . . . . . . . . 15
β’ ((π· β DirRel β§ (π’ β π β§ π£ β π)) β ((π₯ β ((tailβπ·)βπ’) β§ π₯ β ((tailβπ·)βπ£)) β (π’π·π₯ β§ π£π·π₯))) |
64 | 63 | adantr 482 |
. . . . . . . . . . . . . 14
β’ (((π· β DirRel β§ (π’ β π β§ π£ β π)) β§ (π€ β π β§ (π’π·π€ β§ π£π·π€))) β ((π₯ β ((tailβπ·)βπ’) β§ π₯ β ((tailβπ·)βπ£)) β (π’π·π₯ β§ π£π·π₯))) |
65 | 52, 56, 64 | 3imtr4d 294 |
. . . . . . . . . . . . 13
β’ (((π· β DirRel β§ (π’ β π β§ π£ β π)) β§ (π€ β π β§ (π’π·π€ β§ π£π·π€))) β (π₯ β ((tailβπ·)βπ€) β (π₯ β ((tailβπ·)βπ’) β§ π₯ β ((tailβπ·)βπ£)))) |
66 | | elin 3965 |
. . . . . . . . . . . . 13
β’ (π₯ β (((tailβπ·)βπ’) β© ((tailβπ·)βπ£)) β (π₯ β ((tailβπ·)βπ’) β§ π₯ β ((tailβπ·)βπ£))) |
67 | 65, 66 | syl6ibr 252 |
. . . . . . . . . . . 12
β’ (((π· β DirRel β§ (π’ β π β§ π£ β π)) β§ (π€ β π β§ (π’π·π€ β§ π£π·π€))) β (π₯ β ((tailβπ·)βπ€) β π₯ β (((tailβπ·)βπ’) β© ((tailβπ·)βπ£)))) |
68 | 67 | ssrdv 3989 |
. . . . . . . . . . 11
β’ (((π· β DirRel β§ (π’ β π β§ π£ β π)) β§ (π€ β π β§ (π’π·π€ β§ π£π·π€))) β ((tailβπ·)βπ€) β (((tailβπ·)βπ’) β© ((tailβπ·)βπ£))) |
69 | | sseq1 4008 |
. . . . . . . . . . . 12
β’ (π§ = ((tailβπ·)βπ€) β (π§ β (((tailβπ·)βπ’) β© ((tailβπ·)βπ£)) β ((tailβπ·)βπ€) β (((tailβπ·)βπ’) β© ((tailβπ·)βπ£)))) |
70 | 69 | rspcev 3613 |
. . . . . . . . . . 11
β’
((((tailβπ·)βπ€) β ran (tailβπ·) β§ ((tailβπ·)βπ€) β (((tailβπ·)βπ’) β© ((tailβπ·)βπ£))) β βπ§ β ran (tailβπ·)π§ β (((tailβπ·)βπ’) β© ((tailβπ·)βπ£))) |
71 | 36, 68, 70 | syl2anc 585 |
. . . . . . . . . 10
β’ (((π· β DirRel β§ (π’ β π β§ π£ β π)) β§ (π€ β π β§ (π’π·π€ β§ π£π·π€))) β βπ§ β ran (tailβπ·)π§ β (((tailβπ·)βπ’) β© ((tailβπ·)βπ£))) |
72 | 32, 71 | rexlimddv 3162 |
. . . . . . . . 9
β’ ((π· β DirRel β§ (π’ β π β§ π£ β π)) β βπ§ β ran (tailβπ·)π§ β (((tailβπ·)βπ’) β© ((tailβπ·)βπ£))) |
73 | | ineq1 4206 |
. . . . . . . . . . . 12
β’
(((tailβπ·)βπ’) = π₯ β (((tailβπ·)βπ’) β© ((tailβπ·)βπ£)) = (π₯ β© ((tailβπ·)βπ£))) |
74 | 73 | sseq2d 4015 |
. . . . . . . . . . 11
β’
(((tailβπ·)βπ’) = π₯ β (π§ β (((tailβπ·)βπ’) β© ((tailβπ·)βπ£)) β π§ β (π₯ β© ((tailβπ·)βπ£)))) |
75 | 74 | rexbidv 3179 |
. . . . . . . . . 10
β’
(((tailβπ·)βπ’) = π₯ β (βπ§ β ran (tailβπ·)π§ β (((tailβπ·)βπ’) β© ((tailβπ·)βπ£)) β βπ§ β ran (tailβπ·)π§ β (π₯ β© ((tailβπ·)βπ£)))) |
76 | | ineq2 4207 |
. . . . . . . . . . . 12
β’
(((tailβπ·)βπ£) = π¦ β (π₯ β© ((tailβπ·)βπ£)) = (π₯ β© π¦)) |
77 | 76 | sseq2d 4015 |
. . . . . . . . . . 11
β’
(((tailβπ·)βπ£) = π¦ β (π§ β (π₯ β© ((tailβπ·)βπ£)) β π§ β (π₯ β© π¦))) |
78 | 77 | rexbidv 3179 |
. . . . . . . . . 10
β’
(((tailβπ·)βπ£) = π¦ β (βπ§ β ran (tailβπ·)π§ β (π₯ β© ((tailβπ·)βπ£)) β βπ§ β ran (tailβπ·)π§ β (π₯ β© π¦))) |
79 | 75, 78 | sylan9bb 511 |
. . . . . . . . 9
β’
((((tailβπ·)βπ’) = π₯ β§ ((tailβπ·)βπ£) = π¦) β (βπ§ β ran (tailβπ·)π§ β (((tailβπ·)βπ’) β© ((tailβπ·)βπ£)) β βπ§ β ran (tailβπ·)π§ β (π₯ β© π¦))) |
80 | 72, 79 | syl5ibcom 244 |
. . . . . . . 8
β’ ((π· β DirRel β§ (π’ β π β§ π£ β π)) β ((((tailβπ·)βπ’) = π₯ β§ ((tailβπ·)βπ£) = π¦) β βπ§ β ran (tailβπ·)π§ β (π₯ β© π¦))) |
81 | 80 | rexlimdvva 3212 |
. . . . . . 7
β’ (π· β DirRel β
(βπ’ β π βπ£ β π (((tailβπ·)βπ’) = π₯ β§ ((tailβπ·)βπ£) = π¦) β βπ§ β ran (tailβπ·)π§ β (π₯ β© π¦))) |
82 | 30, 81 | biimtrrid 242 |
. . . . . 6
β’ (π· β DirRel β
((βπ’ β π ((tailβπ·)βπ’) = π₯ β§ βπ£ β π ((tailβπ·)βπ£) = π¦) β βπ§ β ran (tailβπ·)π§ β (π₯ β© π¦))) |
83 | 29, 82 | sylbid 239 |
. . . . 5
β’ (π· β DirRel β ((π₯ β ran (tailβπ·) β§ π¦ β ran (tailβπ·)) β βπ§ β ran (tailβπ·)π§ β (π₯ β© π¦))) |
84 | 83 | adantr 482 |
. . . 4
β’ ((π· β DirRel β§ π β β
) β ((π₯ β ran (tailβπ·) β§ π¦ β ran (tailβπ·)) β βπ§ β ran (tailβπ·)π§ β (π₯ β© π¦))) |
85 | 84 | ralrimivv 3199 |
. . 3
β’ ((π· β DirRel β§ π β β
) β
βπ₯ β ran
(tailβπ·)βπ¦ β ran (tailβπ·)βπ§ β ran (tailβπ·)π§ β (π₯ β© π¦)) |
86 | 14, 25, 85 | 3jca 1129 |
. 2
β’ ((π· β DirRel β§ π β β
) β (ran
(tailβπ·) β β
β§ β
β ran (tailβπ·) β§ βπ₯ β ran (tailβπ·)βπ¦ β ran (tailβπ·)βπ§ β ran (tailβπ·)π§ β (π₯ β© π¦))) |
87 | | dmexg 7894 |
. . . . 5
β’ (π· β DirRel β dom π· β V) |
88 | 1, 87 | eqeltrid 2838 |
. . . 4
β’ (π· β DirRel β π β V) |
89 | 88 | adantr 482 |
. . 3
β’ ((π· β DirRel β§ π β β
) β π β V) |
90 | | isfbas2 23339 |
. . 3
β’ (π β V β (ran
(tailβπ·) β
(fBasβπ) β (ran
(tailβπ·) β
π« π β§ (ran
(tailβπ·) β β
β§ β
β ran (tailβπ·) β§ βπ₯ β ran (tailβπ·)βπ¦ β ran (tailβπ·)βπ§ β ran (tailβπ·)π§ β (π₯ β© π¦))))) |
91 | 89, 90 | syl 17 |
. 2
β’ ((π· β DirRel β§ π β β
) β (ran
(tailβπ·) β
(fBasβπ) β (ran
(tailβπ·) β
π« π β§ (ran
(tailβπ·) β β
β§ β
β ran (tailβπ·) β§ βπ₯ β ran (tailβπ·)βπ¦ β ran (tailβπ·)βπ§ β ran (tailβπ·)π§ β (π₯ β© π¦))))) |
92 | 4, 86, 91 | mpbir2and 712 |
1
β’ ((π· β DirRel β§ π β β
) β ran
(tailβπ·) β
(fBasβπ)) |