Step |
Hyp |
Ref |
Expression |
1 |
|
rngcsect.c |
|- C = ( RngCat ` U ) |
2 |
|
rngcsect.b |
|- B = ( Base ` C ) |
3 |
|
rngcsect.u |
|- ( ph -> U e. V ) |
4 |
|
rngcsect.x |
|- ( ph -> X e. B ) |
5 |
|
rngcsect.y |
|- ( ph -> Y e. B ) |
6 |
|
rngciso.n |
|- I = ( Iso ` C ) |
7 |
|
eqid |
|- ( Inv ` C ) = ( Inv ` C ) |
8 |
1
|
rngccat |
|- ( U e. V -> C e. Cat ) |
9 |
3 8
|
syl |
|- ( ph -> C e. Cat ) |
10 |
2 7 9 4 5 6
|
isoval |
|- ( ph -> ( X I Y ) = dom ( X ( Inv ` C ) Y ) ) |
11 |
10
|
eleq2d |
|- ( ph -> ( F e. ( X I Y ) <-> F e. dom ( X ( Inv ` C ) Y ) ) ) |
12 |
2 7 9 4 5
|
invfun |
|- ( ph -> Fun ( X ( Inv ` C ) Y ) ) |
13 |
|
funfvbrb |
|- ( Fun ( X ( Inv ` C ) Y ) -> ( F e. dom ( X ( Inv ` C ) Y ) <-> F ( X ( Inv ` C ) Y ) ( ( X ( Inv ` C ) Y ) ` F ) ) ) |
14 |
12 13
|
syl |
|- ( ph -> ( F e. dom ( X ( Inv ` C ) Y ) <-> F ( X ( Inv ` C ) Y ) ( ( X ( Inv ` C ) Y ) ` F ) ) ) |
15 |
1 2 3 4 5 7
|
rngcinv |
|- ( ph -> ( F ( X ( Inv ` C ) Y ) ( ( X ( Inv ` C ) Y ) ` F ) <-> ( F e. ( X RngIsom Y ) /\ ( ( X ( Inv ` C ) Y ) ` F ) = `' F ) ) ) |
16 |
|
simpl |
|- ( ( F e. ( X RngIsom Y ) /\ ( ( X ( Inv ` C ) Y ) ` F ) = `' F ) -> F e. ( X RngIsom Y ) ) |
17 |
15 16
|
syl6bi |
|- ( ph -> ( F ( X ( Inv ` C ) Y ) ( ( X ( Inv ` C ) Y ) ` F ) -> F e. ( X RngIsom Y ) ) ) |
18 |
14 17
|
sylbid |
|- ( ph -> ( F e. dom ( X ( Inv ` C ) Y ) -> F e. ( X RngIsom Y ) ) ) |
19 |
|
eqid |
|- `' F = `' F |
20 |
1 2 3 4 5 7
|
rngcinv |
|- ( ph -> ( F ( X ( Inv ` C ) Y ) `' F <-> ( F e. ( X RngIsom Y ) /\ `' F = `' F ) ) ) |
21 |
|
funrel |
|- ( Fun ( X ( Inv ` C ) Y ) -> Rel ( X ( Inv ` C ) Y ) ) |
22 |
12 21
|
syl |
|- ( ph -> Rel ( X ( Inv ` C ) Y ) ) |
23 |
|
releldm |
|- ( ( Rel ( X ( Inv ` C ) Y ) /\ F ( X ( Inv ` C ) Y ) `' F ) -> F e. dom ( X ( Inv ` C ) Y ) ) |
24 |
23
|
ex |
|- ( Rel ( X ( Inv ` C ) Y ) -> ( F ( X ( Inv ` C ) Y ) `' F -> F e. dom ( X ( Inv ` C ) Y ) ) ) |
25 |
22 24
|
syl |
|- ( ph -> ( F ( X ( Inv ` C ) Y ) `' F -> F e. dom ( X ( Inv ` C ) Y ) ) ) |
26 |
20 25
|
sylbird |
|- ( ph -> ( ( F e. ( X RngIsom Y ) /\ `' F = `' F ) -> F e. dom ( X ( Inv ` C ) Y ) ) ) |
27 |
19 26
|
mpan2i |
|- ( ph -> ( F e. ( X RngIsom Y ) -> F e. dom ( X ( Inv ` C ) Y ) ) ) |
28 |
18 27
|
impbid |
|- ( ph -> ( F e. dom ( X ( Inv ` C ) Y ) <-> F e. ( X RngIsom Y ) ) ) |
29 |
11 28
|
bitrd |
|- ( ph -> ( F e. ( X I Y ) <-> F e. ( X RngIsom Y ) ) ) |