theorem dmreslam {x: nat} (A: set x) (a: nat x): $ Dom ((\ x, a) |` A) == A $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqstr |
Dom ((\ x, a) |` A) == Dom (\ x, a) i^i A -> Dom (\ x, a) i^i A == A -> Dom ((\ x, a) |` A) == A |
| 2 |
|
dmres |
Dom ((\ x, a) |` A) == Dom (\ x, a) i^i A |
| 3 |
1, 2 |
ax_mp |
Dom (\ x, a) i^i A == A -> Dom ((\ x, a) |` A) == A |
| 4 |
|
eqstr |
Dom (\ x, a) i^i A == _V i^i A -> _V i^i A == A -> Dom (\ x, a) i^i A == A |
| 5 |
|
ineq1 |
Dom (\ x, a) == _V -> Dom (\ x, a) i^i A == _V i^i A |
| 6 |
|
dmlam |
Dom (\ x, a) == _V |
| 7 |
5, 6 |
ax_mp |
Dom (\ x, a) i^i A == _V i^i A |
| 8 |
4, 7 |
ax_mp |
_V i^i A == A -> Dom (\ x, a) i^i A == A |
| 9 |
|
inv1 |
_V i^i A == A |
| 10 |
8, 9 |
ax_mp |
Dom (\ x, a) i^i A == A |
| 11 |
3, 10 |
ax_mp |
Dom ((\ x, a) |` A) == 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
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)