Step |
Hyp |
Ref |
Expression |
1 |
|
fveq2 |
|- ( n = (/) -> ( R1 ` n ) = ( R1 ` (/) ) ) |
2 |
1
|
eleq1d |
|- ( n = (/) -> ( ( R1 ` n ) e. Fin <-> ( R1 ` (/) ) e. Fin ) ) |
3 |
|
fveq2 |
|- ( n = m -> ( R1 ` n ) = ( R1 ` m ) ) |
4 |
3
|
eleq1d |
|- ( n = m -> ( ( R1 ` n ) e. Fin <-> ( R1 ` m ) e. Fin ) ) |
5 |
|
fveq2 |
|- ( n = suc m -> ( R1 ` n ) = ( R1 ` suc m ) ) |
6 |
5
|
eleq1d |
|- ( n = suc m -> ( ( R1 ` n ) e. Fin <-> ( R1 ` suc m ) e. Fin ) ) |
7 |
|
fveq2 |
|- ( n = A -> ( R1 ` n ) = ( R1 ` A ) ) |
8 |
7
|
eleq1d |
|- ( n = A -> ( ( R1 ` n ) e. Fin <-> ( R1 ` A ) e. Fin ) ) |
9 |
|
r10 |
|- ( R1 ` (/) ) = (/) |
10 |
|
0fin |
|- (/) e. Fin |
11 |
9 10
|
eqeltri |
|- ( R1 ` (/) ) e. Fin |
12 |
|
pwfi |
|- ( ( R1 ` m ) e. Fin <-> ~P ( R1 ` m ) e. Fin ) |
13 |
|
r1funlim |
|- ( Fun R1 /\ Lim dom R1 ) |
14 |
13
|
simpri |
|- Lim dom R1 |
15 |
|
limomss |
|- ( Lim dom R1 -> _om C_ dom R1 ) |
16 |
14 15
|
ax-mp |
|- _om C_ dom R1 |
17 |
16
|
sseli |
|- ( m e. _om -> m e. dom R1 ) |
18 |
|
r1sucg |
|- ( m e. dom R1 -> ( R1 ` suc m ) = ~P ( R1 ` m ) ) |
19 |
17 18
|
syl |
|- ( m e. _om -> ( R1 ` suc m ) = ~P ( R1 ` m ) ) |
20 |
19
|
eleq1d |
|- ( m e. _om -> ( ( R1 ` suc m ) e. Fin <-> ~P ( R1 ` m ) e. Fin ) ) |
21 |
12 20
|
bitr4id |
|- ( m e. _om -> ( ( R1 ` m ) e. Fin <-> ( R1 ` suc m ) e. Fin ) ) |
22 |
21
|
biimpd |
|- ( m e. _om -> ( ( R1 ` m ) e. Fin -> ( R1 ` suc m ) e. Fin ) ) |
23 |
2 4 6 8 11 22
|
finds |
|- ( A e. _om -> ( R1 ` A ) e. Fin ) |