theorem elArrow (A B: set) (f: nat): $ f e. Arrow A B <-> func f A B $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
id |
_1 = f -> _1 = f |
| 2 |
1 |
nseqd |
_1 = f -> _1 == f |
| 3 |
2 |
isfeqd |
_1 = f -> (isfun _1 <-> isfun f) |
| 4 |
2 |
dmeqd |
_1 = f -> Dom _1 == Dom f |
| 5 |
4 |
eqseq1d |
_1 = f -> (Dom _1 == A <-> Dom f == A) |
| 6 |
3, 5 |
aneqd |
_1 = f -> (isfun _1 /\ Dom _1 == A <-> isfun f /\ Dom f == A) |
| 7 |
2 |
rneqd |
_1 = f -> Ran _1 == Ran f |
| 8 |
7 |
sseq1d |
_1 = f -> (Ran _1 C_ B <-> Ran f C_ B) |
| 9 |
6, 8 |
aneqd |
_1 = f -> (isfun _1 /\ Dom _1 == A /\ Ran _1 C_ B <-> isfun f /\ Dom f == A /\ Ran f C_ B) |
| 10 |
9 |
conv func |
_1 = f -> (isfun _1 /\ Dom _1 == A /\ Ran _1 C_ B <-> func f A B) |
| 11 |
10 |
elabe |
f e. {_1 | isfun _1 /\ Dom _1 == A /\ Ran _1 C_ B} <-> func f A B |
| 12 |
11 |
conv Arrow |
f e. Arrow A B <-> func 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)