theorem Arrowssxp (A B: set): $ Arrow A B C_ Power (Xp A B) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
elPower |
a1 e. Power (Xp A B) <-> a1 C_ Xp A B |
| 2 |
|
elArrow |
a1 e. Arrow A B <-> func a1 A B |
| 3 |
|
funcssxp |
func a1 A B -> a1 C_ Xp A B |
| 4 |
2, 3 |
sylbi |
a1 e. Arrow A B -> a1 C_ Xp A B |
| 5 |
1, 4 |
sylibr |
a1 e. Arrow A B -> a1 e. Power (Xp A B) |
| 6 |
5 |
ax_gen |
A. a1 (a1 e. Arrow A B -> a1 e. Power (Xp A B)) |
| 7 |
6 |
conv subset |
Arrow A B C_ Power (Xp A B) |
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)