theorem rlameqs {x: nat} (a b: nat x): $ \. x e. a, b == (\ x, b) |` a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqscom |
(\ x, b) |` a == \. x e. a, b -> \. x e. a, b == (\ x, b) |` a |
| 2 |
|
eqlower |
finite ((\ x, b) |` a) <-> (\ x, b) |` a == lower ((\ x, b) |` a) |
| 3 |
2 |
conv rlam |
finite ((\ x, b) |` a) <-> (\ x, b) |` a == \. x e. a, b |
| 4 |
|
finlamh |
finite a -> finite ((\ x, b) |` a) |
| 5 |
|
finns |
finite a |
| 6 |
4, 5 |
ax_mp |
finite ((\ x, b) |` a) |
| 7 |
3, 6 |
mpbi |
(\ x, b) |` a == \. x e. a, b |
| 8 |
1, 7 |
ax_mp |
\. x e. a, b == (\ x, b) |` 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)