theorem appelrn (F: set) (a: nat): $ isfun F -> a e. Dom F -> F @ a e. Ran F $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eldm |
a e. Dom F <-> E. a1 a, a1 e. F |
| 2 |
|
anl |
isfun F /\ a, a1 e. F -> isfun F |
| 3 |
|
anr |
isfun F /\ a, a1 e. F -> a, a1 e. F |
| 4 |
2, 3 |
isfappd |
isfun F /\ a, a1 e. F -> F @ a = a1 |
| 5 |
4 |
eleq1d |
isfun F /\ a, a1 e. F -> (F @ a e. Ran F <-> a1 e. Ran F) |
| 6 |
|
prelrn |
a, a1 e. F -> a1 e. Ran F |
| 7 |
6 |
anwr |
isfun F /\ a, a1 e. F -> a1 e. Ran F |
| 8 |
5, 7 |
mpbird |
isfun F /\ a, a1 e. F -> F @ a e. Ran F |
| 9 |
8 |
eexda |
isfun F -> E. a1 a, a1 e. F -> F @ a e. Ran F |
| 10 |
1, 9 |
syl5bi |
isfun F -> a e. Dom F -> F @ a e. Ran 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
(peano2,
addeq,
muleq)