theorem finlamh {x: nat} (A: set x) (v: nat x):
$ finite A -> finite ((\ x, v) |` A) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
fineq |
(\ x, v) |` A == (\ a1, N[a1 / x] v) |` A -> (finite ((\ x, v) |` A) <-> finite ((\ a1, N[a1 / x] v) |` A)) |
| 2 |
|
reseq1 |
\ x, v == \ a1, N[a1 / x] v -> (\ x, v) |` A == (\ a1, N[a1 / x] v) |` A |
| 3 |
|
cbvlams |
\ x, v == \ a1, N[a1 / x] v |
| 4 |
2, 3 |
ax_mp |
(\ x, v) |` A == (\ a1, N[a1 / x] v) |` A |
| 5 |
1, 4 |
ax_mp |
finite ((\ x, v) |` A) <-> finite ((\ a1, N[a1 / x] v) |` A) |
| 6 |
|
finlam |
finite A -> finite ((\ a1, N[a1 / x] v) |` A) |
| 7 |
5, 6 |
sylibr |
finite A -> finite ((\ x, v) |` A) |
Axiom use
axs_prop_calc
(ax_1,
ax_2,
ax_3,
ax_mp,
itru),
axs_pred_calc
(ax_gen,
ax_4,
ax_5,
ax_6,
ax_7,
ax_10,
ax_11,
ax_12),
axs_set
(elab,
ax_8),
axs_the
(theid,
the0),
axs_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)