Step |
Hyp |
Ref |
Expression |
1 |
|
fof |
|- ( f : A -onto-> B -> f : A --> B ) |
2 |
1
|
fdmd |
|- ( f : A -onto-> B -> dom f = A ) |
3 |
2
|
eqeq1d |
|- ( f : A -onto-> B -> ( dom f = (/) <-> A = (/) ) ) |
4 |
|
dm0rn0 |
|- ( dom f = (/) <-> ran f = (/) ) |
5 |
|
forn |
|- ( f : A -onto-> B -> ran f = B ) |
6 |
5
|
eqeq1d |
|- ( f : A -onto-> B -> ( ran f = (/) <-> B = (/) ) ) |
7 |
4 6
|
syl5bb |
|- ( f : A -onto-> B -> ( dom f = (/) <-> B = (/) ) ) |
8 |
3 7
|
bitr3d |
|- ( f : A -onto-> B -> ( A = (/) <-> B = (/) ) ) |
9 |
8
|
necon3bid |
|- ( f : A -onto-> B -> ( A =/= (/) <-> B =/= (/) ) ) |
10 |
9
|
biimpac |
|- ( ( A =/= (/) /\ f : A -onto-> B ) -> B =/= (/) ) |
11 |
|
vex |
|- f e. _V |
12 |
11
|
dmex |
|- dom f e. _V |
13 |
2 12
|
eqeltrrdi |
|- ( f : A -onto-> B -> A e. _V ) |
14 |
|
fornex |
|- ( A e. _V -> ( f : A -onto-> B -> B e. _V ) ) |
15 |
13 14
|
mpcom |
|- ( f : A -onto-> B -> B e. _V ) |
16 |
|
0sdomg |
|- ( B e. _V -> ( (/) ~< B <-> B =/= (/) ) ) |
17 |
15 16
|
syl |
|- ( f : A -onto-> B -> ( (/) ~< B <-> B =/= (/) ) ) |
18 |
17
|
adantl |
|- ( ( A =/= (/) /\ f : A -onto-> B ) -> ( (/) ~< B <-> B =/= (/) ) ) |
19 |
10 18
|
mpbird |
|- ( ( A =/= (/) /\ f : A -onto-> B ) -> (/) ~< B ) |
20 |
19
|
ex |
|- ( A =/= (/) -> ( f : A -onto-> B -> (/) ~< B ) ) |
21 |
|
fodomg |
|- ( A e. _V -> ( f : A -onto-> B -> B ~<_ A ) ) |
22 |
13 21
|
mpcom |
|- ( f : A -onto-> B -> B ~<_ A ) |
23 |
20 22
|
jca2 |
|- ( A =/= (/) -> ( f : A -onto-> B -> ( (/) ~< B /\ B ~<_ A ) ) ) |
24 |
23
|
exlimdv |
|- ( A =/= (/) -> ( E. f f : A -onto-> B -> ( (/) ~< B /\ B ~<_ A ) ) ) |
25 |
24
|
imp |
|- ( ( A =/= (/) /\ E. f f : A -onto-> B ) -> ( (/) ~< B /\ B ~<_ A ) ) |
26 |
|
sdomdomtr |
|- ( ( (/) ~< B /\ B ~<_ A ) -> (/) ~< A ) |
27 |
|
reldom |
|- Rel ~<_ |
28 |
27
|
brrelex2i |
|- ( B ~<_ A -> A e. _V ) |
29 |
28
|
adantl |
|- ( ( (/) ~< B /\ B ~<_ A ) -> A e. _V ) |
30 |
|
0sdomg |
|- ( A e. _V -> ( (/) ~< A <-> A =/= (/) ) ) |
31 |
29 30
|
syl |
|- ( ( (/) ~< B /\ B ~<_ A ) -> ( (/) ~< A <-> A =/= (/) ) ) |
32 |
26 31
|
mpbid |
|- ( ( (/) ~< B /\ B ~<_ A ) -> A =/= (/) ) |
33 |
|
fodomr |
|- ( ( (/) ~< B /\ B ~<_ A ) -> E. f f : A -onto-> B ) |
34 |
32 33
|
jca |
|- ( ( (/) ~< B /\ B ~<_ A ) -> ( A =/= (/) /\ E. f f : A -onto-> B ) ) |
35 |
25 34
|
impbii |
|- ( ( A =/= (/) /\ E. f f : A -onto-> B ) <-> ( (/) ~< B /\ B ~<_ A ) ) |