theorem sapprapp (F: set) (a b: nat): $ F @@ a @' b == F @' (a, b) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqidd |
_1 = b -> y = y |
| 2 |
|
eqsidd |
_1 = b -> F == F |
| 3 |
|
eqidd |
_1 = b -> a = a |
| 4 |
|
id |
_1 = b -> _1 = b |
| 5 |
3, 4 |
preqd |
_1 = b -> a, _1 = a, b |
| 6 |
2, 5 |
rappeqd |
_1 = b -> F @' (a, _1) == F @' (a, b) |
| 7 |
1, 6 |
eleqd |
_1 = b -> (y e. F @' (a, _1) <-> y e. F @' (a, b)) |
| 8 |
7 |
elsabe |
b, y e. S\ _1, F @' (a, _1) <-> y e. F @' (a, b) |
| 9 |
8 |
conv sapp |
b, y e. F @@ a <-> y e. F @' (a, b) |
| 10 |
9 |
eqab1i |
{y | b, y e. F @@ a} == F @' (a, b) |
| 11 |
10 |
conv rapp |
F @@ a @' b == F @' (a, b) |
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)