| Step |
Hyp |
Ref |
Expression |
| 1 |
|
tpf1o.f |
|- F = ( x e. ( 0 ..^ 3 ) |-> if ( x = 0 , A , if ( x = 1 , B , C ) ) ) |
| 2 |
|
tpf.t |
|- T = { A , B , C } |
| 3 |
1 2
|
tpfo |
|- ( ( A e. V /\ B e. V /\ C e. V ) -> F : ( 0 ..^ 3 ) -onto-> T ) |
| 4 |
3
|
adantr |
|- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( # ` T ) = 3 ) -> F : ( 0 ..^ 3 ) -onto-> T ) |
| 5 |
|
3nn0 |
|- 3 e. NN0 |
| 6 |
|
hashfzo0 |
|- ( 3 e. NN0 -> ( # ` ( 0 ..^ 3 ) ) = 3 ) |
| 7 |
5 6
|
ax-mp |
|- ( # ` ( 0 ..^ 3 ) ) = 3 |
| 8 |
|
eqcom |
|- ( ( # ` T ) = 3 <-> 3 = ( # ` T ) ) |
| 9 |
8
|
biimpi |
|- ( ( # ` T ) = 3 -> 3 = ( # ` T ) ) |
| 10 |
9
|
adantl |
|- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( # ` T ) = 3 ) -> 3 = ( # ` T ) ) |
| 11 |
7 10
|
eqtrid |
|- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( # ` T ) = 3 ) -> ( # ` ( 0 ..^ 3 ) ) = ( # ` T ) ) |
| 12 |
|
fzofi |
|- ( 0 ..^ 3 ) e. Fin |
| 13 |
12
|
a1i |
|- ( ( A e. V /\ B e. V /\ C e. V ) -> ( 0 ..^ 3 ) e. Fin ) |
| 14 |
|
tpfi |
|- { A , B , C } e. Fin |
| 15 |
2 14
|
eqeltri |
|- T e. Fin |
| 16 |
15
|
a1i |
|- ( ( # ` T ) = 3 -> T e. Fin ) |
| 17 |
|
hashen |
|- ( ( ( 0 ..^ 3 ) e. Fin /\ T e. Fin ) -> ( ( # ` ( 0 ..^ 3 ) ) = ( # ` T ) <-> ( 0 ..^ 3 ) ~~ T ) ) |
| 18 |
13 16 17
|
syl2an |
|- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( # ` T ) = 3 ) -> ( ( # ` ( 0 ..^ 3 ) ) = ( # ` T ) <-> ( 0 ..^ 3 ) ~~ T ) ) |
| 19 |
11 18
|
mpbid |
|- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( # ` T ) = 3 ) -> ( 0 ..^ 3 ) ~~ T ) |
| 20 |
15
|
a1i |
|- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( # ` T ) = 3 ) -> T e. Fin ) |
| 21 |
|
fofinf1o |
|- ( ( F : ( 0 ..^ 3 ) -onto-> T /\ ( 0 ..^ 3 ) ~~ T /\ T e. Fin ) -> F : ( 0 ..^ 3 ) -1-1-onto-> T ) |
| 22 |
4 19 20 21
|
syl3anc |
|- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( # ` T ) = 3 ) -> F : ( 0 ..^ 3 ) -1-1-onto-> T ) |