Step |
Hyp |
Ref |
Expression |
1 |
|
funcringcsetcALTV.r |
|- R = ( RingCatALTV ` U ) |
2 |
|
funcringcsetcALTV.s |
|- S = ( SetCat ` U ) |
3 |
|
funcringcsetcALTV.b |
|- B = ( Base ` R ) |
4 |
|
funcringcsetcALTV.c |
|- C = ( Base ` S ) |
5 |
|
funcringcsetcALTV.u |
|- ( ph -> U e. WUni ) |
6 |
|
funcringcsetcALTV.f |
|- ( ph -> F = ( x e. B |-> ( Base ` x ) ) ) |
7 |
1 3 5
|
ringcbasbasALTV |
|- ( ( ph /\ x e. B ) -> ( Base ` x ) e. U ) |
8 |
2 5
|
setcbas |
|- ( ph -> U = ( Base ` S ) ) |
9 |
8
|
eqcomd |
|- ( ph -> ( Base ` S ) = U ) |
10 |
9
|
adantr |
|- ( ( ph /\ x e. B ) -> ( Base ` S ) = U ) |
11 |
7 10
|
eleqtrrd |
|- ( ( ph /\ x e. B ) -> ( Base ` x ) e. ( Base ` S ) ) |
12 |
11 4
|
eleqtrrdi |
|- ( ( ph /\ x e. B ) -> ( Base ` x ) e. C ) |
13 |
12
|
fmpttd |
|- ( ph -> ( x e. B |-> ( Base ` x ) ) : B --> C ) |
14 |
6
|
feq1d |
|- ( ph -> ( F : B --> C <-> ( x e. B |-> ( Base ` x ) ) : B --> C ) ) |
15 |
13 14
|
mpbird |
|- ( ph -> F : B --> C ) |