Step |
Hyp |
Ref |
Expression |
1 |
|
fin1a2lem.a |
|- S = ( x e. On |-> suc x ) |
2 |
|
suceloni |
|- ( x e. On -> suc x e. On ) |
3 |
1 2
|
fmpti |
|- S : On --> On |
4 |
1
|
fin1a2lem1 |
|- ( a e. On -> ( S ` a ) = suc a ) |
5 |
1
|
fin1a2lem1 |
|- ( b e. On -> ( S ` b ) = suc b ) |
6 |
4 5
|
eqeqan12d |
|- ( ( a e. On /\ b e. On ) -> ( ( S ` a ) = ( S ` b ) <-> suc a = suc b ) ) |
7 |
|
suc11 |
|- ( ( a e. On /\ b e. On ) -> ( suc a = suc b <-> a = b ) ) |
8 |
6 7
|
bitrd |
|- ( ( a e. On /\ b e. On ) -> ( ( S ` a ) = ( S ` b ) <-> a = b ) ) |
9 |
8
|
biimpd |
|- ( ( a e. On /\ b e. On ) -> ( ( S ` a ) = ( S ` b ) -> a = b ) ) |
10 |
9
|
rgen2 |
|- A. a e. On A. b e. On ( ( S ` a ) = ( S ` b ) -> a = b ) |
11 |
|
dff13 |
|- ( S : On -1-1-> On <-> ( S : On --> On /\ A. a e. On A. b e. On ( ( S ` a ) = ( S ` b ) -> a = b ) ) ) |
12 |
3 10 11
|
mpbir2an |
|- S : On -1-1-> On |