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 |
|
funcringcsetcALTV.g |
|- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) ) |
8 |
|
eqid |
|- ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) = ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) |
9 |
|
ovex |
|- ( x RingHom y ) e. _V |
10 |
|
id |
|- ( ( x RingHom y ) e. _V -> ( x RingHom y ) e. _V ) |
11 |
10
|
resiexd |
|- ( ( x RingHom y ) e. _V -> ( _I |` ( x RingHom y ) ) e. _V ) |
12 |
9 11
|
ax-mp |
|- ( _I |` ( x RingHom y ) ) e. _V |
13 |
8 12
|
fnmpoi |
|- ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) Fn ( B X. B ) |
14 |
7
|
fneq1d |
|- ( ph -> ( G Fn ( B X. B ) <-> ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) Fn ( B X. B ) ) ) |
15 |
13 14
|
mpbiri |
|- ( ph -> G Fn ( B X. B ) ) |