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 |
|
rngcsect.e |
|- E = ( Base ` X ) |
7 |
|
rngcsect.n |
|- S = ( Sect ` C ) |
8 |
|
eqid |
|- ( Hom ` C ) = ( Hom ` C ) |
9 |
|
eqid |
|- ( comp ` C ) = ( comp ` C ) |
10 |
|
eqid |
|- ( Id ` C ) = ( Id ` C ) |
11 |
1
|
rngccat |
|- ( U e. V -> C e. Cat ) |
12 |
3 11
|
syl |
|- ( ph -> C e. Cat ) |
13 |
2 8 9 10 7 12 4 5
|
issect |
|- ( ph -> ( F ( X S Y ) G <-> ( F e. ( X ( Hom ` C ) Y ) /\ G e. ( Y ( Hom ` C ) X ) /\ ( G ( <. X , Y >. ( comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) ) ) ) |
14 |
1 2 3 8 4 5
|
rngchom |
|- ( ph -> ( X ( Hom ` C ) Y ) = ( X RngHomo Y ) ) |
15 |
14
|
eleq2d |
|- ( ph -> ( F e. ( X ( Hom ` C ) Y ) <-> F e. ( X RngHomo Y ) ) ) |
16 |
1 2 3 8 5 4
|
rngchom |
|- ( ph -> ( Y ( Hom ` C ) X ) = ( Y RngHomo X ) ) |
17 |
16
|
eleq2d |
|- ( ph -> ( G e. ( Y ( Hom ` C ) X ) <-> G e. ( Y RngHomo X ) ) ) |
18 |
15 17
|
anbi12d |
|- ( ph -> ( ( F e. ( X ( Hom ` C ) Y ) /\ G e. ( Y ( Hom ` C ) X ) ) <-> ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) ) ) |
19 |
18
|
anbi1d |
|- ( ph -> ( ( ( F e. ( X ( Hom ` C ) Y ) /\ G e. ( Y ( Hom ` C ) X ) ) /\ ( G ( <. X , Y >. ( comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) ) <-> ( ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) /\ ( G ( <. X , Y >. ( comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) ) ) ) |
20 |
3
|
adantr |
|- ( ( ph /\ ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) ) -> U e. V ) |
21 |
4
|
adantr |
|- ( ( ph /\ ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) ) -> X e. B ) |
22 |
1 2 3
|
rngcbas |
|- ( ph -> B = ( U i^i Rng ) ) |
23 |
22
|
eleq2d |
|- ( ph -> ( X e. B <-> X e. ( U i^i Rng ) ) ) |
24 |
|
inss1 |
|- ( U i^i Rng ) C_ U |
25 |
24
|
a1i |
|- ( ph -> ( U i^i Rng ) C_ U ) |
26 |
25
|
sseld |
|- ( ph -> ( X e. ( U i^i Rng ) -> X e. U ) ) |
27 |
23 26
|
sylbid |
|- ( ph -> ( X e. B -> X e. U ) ) |
28 |
27
|
adantr |
|- ( ( ph /\ ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) ) -> ( X e. B -> X e. U ) ) |
29 |
21 28
|
mpd |
|- ( ( ph /\ ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) ) -> X e. U ) |
30 |
5
|
adantr |
|- ( ( ph /\ ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) ) -> Y e. B ) |
31 |
22
|
eleq2d |
|- ( ph -> ( Y e. B <-> Y e. ( U i^i Rng ) ) ) |
32 |
25
|
sseld |
|- ( ph -> ( Y e. ( U i^i Rng ) -> Y e. U ) ) |
33 |
31 32
|
sylbid |
|- ( ph -> ( Y e. B -> Y e. U ) ) |
34 |
33
|
adantr |
|- ( ( ph /\ ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) ) -> ( Y e. B -> Y e. U ) ) |
35 |
30 34
|
mpd |
|- ( ( ph /\ ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) ) -> Y e. U ) |
36 |
|
eqid |
|- ( Base ` X ) = ( Base ` X ) |
37 |
|
eqid |
|- ( Base ` Y ) = ( Base ` Y ) |
38 |
36 37
|
rnghmf |
|- ( F e. ( X RngHomo Y ) -> F : ( Base ` X ) --> ( Base ` Y ) ) |
39 |
38
|
adantr |
|- ( ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) -> F : ( Base ` X ) --> ( Base ` Y ) ) |
40 |
39
|
adantl |
|- ( ( ph /\ ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) ) -> F : ( Base ` X ) --> ( Base ` Y ) ) |
41 |
37 36
|
rnghmf |
|- ( G e. ( Y RngHomo X ) -> G : ( Base ` Y ) --> ( Base ` X ) ) |
42 |
41
|
adantl |
|- ( ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) -> G : ( Base ` Y ) --> ( Base ` X ) ) |
43 |
42
|
adantl |
|- ( ( ph /\ ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) ) -> G : ( Base ` Y ) --> ( Base ` X ) ) |
44 |
1 20 9 29 35 29 40 43
|
rngcco |
|- ( ( ph /\ ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) ) -> ( G ( <. X , Y >. ( comp ` C ) X ) F ) = ( G o. F ) ) |
45 |
1 2 10 3 4 6
|
rngcid |
|- ( ph -> ( ( Id ` C ) ` X ) = ( _I |` E ) ) |
46 |
45
|
adantr |
|- ( ( ph /\ ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) ) -> ( ( Id ` C ) ` X ) = ( _I |` E ) ) |
47 |
44 46
|
eqeq12d |
|- ( ( ph /\ ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) ) -> ( ( G ( <. X , Y >. ( comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) <-> ( G o. F ) = ( _I |` E ) ) ) |
48 |
47
|
pm5.32da |
|- ( ph -> ( ( ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) /\ ( G ( <. X , Y >. ( comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) ) <-> ( ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) /\ ( G o. F ) = ( _I |` E ) ) ) ) |
49 |
19 48
|
bitrd |
|- ( ph -> ( ( ( F e. ( X ( Hom ` C ) Y ) /\ G e. ( Y ( Hom ` C ) X ) ) /\ ( G ( <. X , Y >. ( comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) ) <-> ( ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) /\ ( G o. F ) = ( _I |` E ) ) ) ) |
50 |
|
df-3an |
|- ( ( F e. ( X ( Hom ` C ) Y ) /\ G e. ( Y ( Hom ` C ) X ) /\ ( G ( <. X , Y >. ( comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) ) <-> ( ( F e. ( X ( Hom ` C ) Y ) /\ G e. ( Y ( Hom ` C ) X ) ) /\ ( G ( <. X , Y >. ( comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) ) ) |
51 |
|
df-3an |
|- ( ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) /\ ( G o. F ) = ( _I |` E ) ) <-> ( ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) ) /\ ( G o. F ) = ( _I |` E ) ) ) |
52 |
49 50 51
|
3bitr4g |
|- ( ph -> ( ( F e. ( X ( Hom ` C ) Y ) /\ G e. ( Y ( Hom ` C ) X ) /\ ( G ( <. X , Y >. ( comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) ) <-> ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) /\ ( G o. F ) = ( _I |` E ) ) ) ) |
53 |
13 52
|
bitrd |
|- ( ph -> ( F ( X S Y ) G <-> ( F e. ( X RngHomo Y ) /\ G e. ( Y RngHomo X ) /\ ( G o. F ) = ( _I |` E ) ) ) ) |