Step | Hyp | Ref
| Expression |
1 | | df-inr 9846 |
. . 3
ā¢ inr =
(š„ ā V ā¦
āØ1o, š„ā©) |
2 | | 1onn 8591 |
. . . . . 6
ā¢
1o ā Ļ |
3 | | snidg 4625 |
. . . . . 6
ā¢
(1o ā Ļ ā 1o ā
{1o}) |
4 | 2, 3 | ax-mp 5 |
. . . . 5
ā¢
1o ā {1o} |
5 | | opelxpi 5675 |
. . . . 5
ā¢
((1o ā {1o} ā§ š„ ā V) ā āØ1o, š„ā© ā ({1o}
Ć V)) |
6 | 4, 5 | mpan 689 |
. . . 4
ā¢ (š„ ā V ā
āØ1o, š„ā© ā ({1o} Ć
V)) |
7 | 6 | adantl 483 |
. . 3
ā¢
((ā¤ ā§ š„
ā V) ā āØ1o, š„ā© ā ({1o} Ć
V)) |
8 | | fvexd 6862 |
. . 3
ā¢
((ā¤ ā§ š¦
ā ({1o} Ć V)) ā (2nd āš¦) ā V) |
9 | | 1st2nd2 7965 |
. . . . . . . 8
ā¢ (š¦ ā ({1o} Ć
V) ā š¦ =
āØ(1st āš¦), (2nd āš¦)ā©) |
10 | | xp1st 7958 |
. . . . . . . . . 10
ā¢ (š¦ ā ({1o} Ć
V) ā (1st āš¦) ā {1o}) |
11 | | elsni 4608 |
. . . . . . . . . 10
ā¢
((1st āš¦) ā {1o} ā
(1st āš¦) =
1o) |
12 | 10, 11 | syl 17 |
. . . . . . . . 9
ā¢ (š¦ ā ({1o} Ć
V) ā (1st āš¦) = 1o) |
13 | 12 | opeq1d 4841 |
. . . . . . . 8
ā¢ (š¦ ā ({1o} Ć
V) ā āØ(1st āš¦), (2nd āš¦)ā© = āØ1o,
(2nd āš¦)ā©) |
14 | 9, 13 | eqtrd 2777 |
. . . . . . 7
ā¢ (š¦ ā ({1o} Ć
V) ā š¦ =
āØ1o, (2nd āš¦)ā©) |
15 | 14 | eqeq2d 2748 |
. . . . . 6
ā¢ (š¦ ā ({1o} Ć
V) ā (āØ1o, š„ā© = š¦ ā āØ1o, š„ā© = āØ1o,
(2nd āš¦)ā©)) |
16 | | eqcom 2744 |
. . . . . 6
ā¢
(āØ1o, š„ā© = š¦ ā š¦ = āØ1o, š„ā©) |
17 | | eqid 2737 |
. . . . . . 7
ā¢
1o = 1o |
18 | | 1oex 8427 |
. . . . . . . 8
ā¢
1o ā V |
19 | | vex 3452 |
. . . . . . . 8
ā¢ š„ ā V |
20 | 18, 19 | opth 5438 |
. . . . . . 7
ā¢
(āØ1o, š„ā© = āØ1o, (2nd
āš¦)ā© ā
(1o = 1o ā§ š„ = (2nd āš¦))) |
21 | 17, 20 | mpbiran 708 |
. . . . . 6
ā¢
(āØ1o, š„ā© = āØ1o, (2nd
āš¦)ā© ā
š„ = (2nd
āš¦)) |
22 | 15, 16, 21 | 3bitr3g 313 |
. . . . 5
ā¢ (š¦ ā ({1o} Ć
V) ā (š¦ =
āØ1o, š„ā© ā š„ = (2nd āš¦))) |
23 | 22 | bicomd 222 |
. . . 4
ā¢ (š¦ ā ({1o} Ć
V) ā (š„ =
(2nd āš¦)
ā š¦ =
āØ1o, š„ā©)) |
24 | 23 | ad2antll 728 |
. . 3
ā¢
((ā¤ ā§ (š„
ā V ā§ š¦ ā
({1o} Ć V))) ā (š„ = (2nd āš¦) ā š¦ = āØ1o, š„ā©)) |
25 | 1, 7, 8, 24 | f1o2d 7612 |
. 2
ā¢ (ā¤
ā inr:Vā1-1-ontoā({1o} Ć V)) |
26 | 25 | mptru 1549 |
1
ā¢
inr:Vā1-1-ontoā({1o} Ć V) |