Metamath Proof Explorer


Theorem funcringcsetclem3ALTV

Description: Lemma 3 for funcringcsetcALTV . (Contributed by AV, 15-Feb-2020) (New usage is discouraged.)

Ref Expression
Hypotheses funcringcsetcALTV.r
|- R = ( RingCatALTV ` U )
funcringcsetcALTV.s
|- S = ( SetCat ` U )
funcringcsetcALTV.b
|- B = ( Base ` R )
funcringcsetcALTV.c
|- C = ( Base ` S )
funcringcsetcALTV.u
|- ( ph -> U e. WUni )
funcringcsetcALTV.f
|- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )
Assertion funcringcsetclem3ALTV
|- ( ph -> F : B --> C )

Proof

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 )