Metamath Proof Explorer


Theorem funcringcsetcALTV2lem4

Description: Lemma 4 for funcringcsetcALTV2 . (Contributed by AV, 15-Feb-2020) (New usage is discouraged.)

Ref Expression
Hypotheses funcringcsetcALTV2.r
|- R = ( RingCat ` U )
funcringcsetcALTV2.s
|- S = ( SetCat ` U )
funcringcsetcALTV2.b
|- B = ( Base ` R )
funcringcsetcALTV2.c
|- C = ( Base ` S )
funcringcsetcALTV2.u
|- ( ph -> U e. WUni )
funcringcsetcALTV2.f
|- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )
funcringcsetcALTV2.g
|- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) )
Assertion funcringcsetcALTV2lem4
|- ( ph -> G Fn ( B X. B ) )

Proof

Step Hyp Ref Expression
1 funcringcsetcALTV2.r
 |-  R = ( RingCat ` U )
2 funcringcsetcALTV2.s
 |-  S = ( SetCat ` U )
3 funcringcsetcALTV2.b
 |-  B = ( Base ` R )
4 funcringcsetcALTV2.c
 |-  C = ( Base ` S )
5 funcringcsetcALTV2.u
 |-  ( ph -> U e. WUni )
6 funcringcsetcALTV2.f
 |-  ( ph -> F = ( x e. B |-> ( Base ` x ) ) )
7 funcringcsetcALTV2.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 ) )