Step |
Hyp |
Ref |
Expression |
1 |
|
invfval.b |
|- B = ( Base ` C ) |
2 |
|
invfval.n |
|- N = ( Inv ` C ) |
3 |
|
invfval.c |
|- ( ph -> C e. Cat ) |
4 |
|
invfval.x |
|- ( ph -> X e. B ) |
5 |
|
invfval.y |
|- ( ph -> Y e. B ) |
6 |
|
isoval.n |
|- I = ( Iso ` C ) |
7 |
1 2 3 4 5
|
invfun |
|- ( ph -> Fun ( X N Y ) ) |
8 |
7
|
funfnd |
|- ( ph -> ( X N Y ) Fn dom ( X N Y ) ) |
9 |
1 2 3 4 5 6
|
isoval |
|- ( ph -> ( X I Y ) = dom ( X N Y ) ) |
10 |
9
|
fneq2d |
|- ( ph -> ( ( X N Y ) Fn ( X I Y ) <-> ( X N Y ) Fn dom ( X N Y ) ) ) |
11 |
8 10
|
mpbird |
|- ( ph -> ( X N Y ) Fn ( X I Y ) ) |
12 |
|
df-rn |
|- ran ( X N Y ) = dom `' ( X N Y ) |
13 |
1 2 3 4 5
|
invsym2 |
|- ( ph -> `' ( X N Y ) = ( Y N X ) ) |
14 |
13
|
dmeqd |
|- ( ph -> dom `' ( X N Y ) = dom ( Y N X ) ) |
15 |
1 2 3 5 4 6
|
isoval |
|- ( ph -> ( Y I X ) = dom ( Y N X ) ) |
16 |
14 15
|
eqtr4d |
|- ( ph -> dom `' ( X N Y ) = ( Y I X ) ) |
17 |
12 16
|
eqtrid |
|- ( ph -> ran ( X N Y ) = ( Y I X ) ) |
18 |
|
eqimss |
|- ( ran ( X N Y ) = ( Y I X ) -> ran ( X N Y ) C_ ( Y I X ) ) |
19 |
17 18
|
syl |
|- ( ph -> ran ( X N Y ) C_ ( Y I X ) ) |
20 |
|
df-f |
|- ( ( X N Y ) : ( X I Y ) --> ( Y I X ) <-> ( ( X N Y ) Fn ( X I Y ) /\ ran ( X N Y ) C_ ( Y I X ) ) ) |
21 |
11 19 20
|
sylanbrc |
|- ( ph -> ( X N Y ) : ( X I Y ) --> ( Y I X ) ) |