theorem isfrn (F: set) {x: nat} (y: nat):
$ isfun F -> (y e. Ran F <-> E. x (x e. Dom F /\ F @ x = y)) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
elrn |
y e. Ran F <-> E. x x, y e. F |
| 2 |
|
isfappb |
isfun F -> (x, y e. F <-> x e. Dom F /\ F @ x = y) |
| 3 |
2 |
exeqd |
isfun F -> (E. x x, y e. F <-> E. x (x e. Dom F /\ F @ x = y)) |
| 4 |
1, 3 |
syl5bb |
isfun F -> (y e. Ran F <-> E. x (x e. Dom F /\ F @ x = y)) |
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)