theorem finlamima (A: set) {x: nat} (v: nat x):
$ finite A -> finite ((\ x, v) '' A) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
fineq |
Ran ((\ x, v) |` A) == (\ x, v) '' A -> (finite (Ran ((\ x, v) |` A)) <-> finite ((\ x, v) '' A)) |
| 2 |
|
rnres |
Ran ((\ x, v) |` A) == (\ x, v) '' A |
| 3 |
1, 2 |
ax_mp |
finite (Ran ((\ x, v) |` A)) <-> finite ((\ x, v) '' A) |
| 4 |
|
rnfin |
finite ((\ x, v) |` A) -> finite (Ran ((\ x, v) |` A)) |
| 5 |
|
finlam |
finite A -> finite ((\ x, v) |` A) |
| 6 |
4, 5 |
syl |
finite A -> finite (Ran ((\ x, v) |` A)) |
| 7 |
3, 6 |
sylib |
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)