theorem funcfin (A B F: set):
$ func F A B -> finite A -> finite B -> finite F $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
xpfin |
finite A -> finite B -> finite (Xp A B) |
| 2 |
|
funcssxp |
func F A B -> F C_ Xp A B |
| 3 |
|
finss |
F C_ Xp A B -> finite (Xp A B) -> finite F |
| 4 |
2, 3 |
rsyl |
func F A B -> finite (Xp A B) -> finite F |
| 5 |
4 |
imim2d |
func F A B -> (finite B -> finite (Xp A B)) -> finite B -> finite F |
| 6 |
1, 5 |
syl5 |
func F A B -> finite A -> finite B -> finite F |
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)