theorem isfappd (F: set) (G: wff) (a b: nat):
$ G -> isfun F $ >
$ G -> a, b e. F $ >
$ G -> F @ a = b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
elrapp |
x e. F @' a <-> a, x e. F |
| 2 |
1 |
conv rapp |
x e. {a1 | a, a1 e. F} <-> a, x e. F |
| 3 |
|
eqcomb |
b = x <-> x = b |
| 4 |
|
hyp h1 |
G -> isfun F |
| 5 |
|
hyp h2 |
G -> a, b e. F |
| 6 |
4, 5 |
isfbd |
G -> (a, x e. F <-> b = x) |
| 7 |
3, 6 |
syl6bb |
G -> (a, x e. F <-> x = b) |
| 8 |
2, 7 |
syl5bb |
G -> (x e. {a1 | a, a1 e. F} <-> x = b) |
| 9 |
8 |
eqthed |
G -> the {a1 | a, a1 e. F} = b |
| 10 |
9 |
conv app |
G -> 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
(peano2,
addeq,
muleq)