theorem eldm (A: set) (a: nat) {y: nat}: $ a e. Dom A <-> E. y a, y e. A $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
preq1 |
x = a -> x, y = a, y |
| 2 |
1 |
eleq1d |
x = a -> (x, y e. A <-> a, y e. A) |
| 3 |
2 |
exeqd |
x = a -> (E. y x, y e. A <-> E. y a, y e. A) |
| 4 |
3 |
elabe |
a e. {x | E. y x, y e. A} <-> E. y a, y e. A |
| 5 |
4 |
conv Dom |
a e. Dom A <-> E. y a, y e. 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
(peano2,
addeq,
muleq)