Theorem elArrow ≪ | index | src | ≫ 

theorem elArrow (A B: set) (f: nat): $ f e. Arrow A B <-> func f A B $;
StepHypRefExpression
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 eqsidd 
_1 = f -> A == A
6 4, 5 eqseqd 
_1 = f -> (Dom _1 == A <-> Dom f == A)
7 3, 6 aneqd 
_1 = f -> (isfun _1 /\ Dom _1 == A <-> isfun f /\ Dom f == A)
8 2 rneqd 
_1 = f -> Ran _1 == Ran f
9 eqsidd 
_1 = f -> B == B
10 8, 9 sseqd 
_1 = f -> (Ran _1 C_ B <-> Ran f C_ B)
11 7, 10 aneqd 
_1 = f -> (isfun _1 /\ Dom _1 == A /\ Ran _1 C_ B <-> isfun f /\ Dom f == A /\ Ran f C_ B)
12 11 conv func 
_1 = f -> (isfun _1 /\ Dom _1 == A /\ Ran _1 C_ B <-> func f A B)
13 12 elabe 
f e. {_1 | isfun _1 /\ Dom _1 == A /\ Ran _1 C_ B} <-> func f A B
14 13 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)