Metamath Proof Explorer


Theorem funcestrcsetclem3

Description: Lemma 3 for funcestrcsetc . (Contributed by AV, 22-Mar-2020)

Ref Expression
Hypotheses funcestrcsetc.e
|- E = ( ExtStrCat ` U )
funcestrcsetc.s
|- S = ( SetCat ` U )
funcestrcsetc.b
|- B = ( Base ` E )
funcestrcsetc.c
|- C = ( Base ` S )
funcestrcsetc.u
|- ( ph -> U e. WUni )
funcestrcsetc.f
|- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )
Assertion funcestrcsetclem3
|- ( ph -> F : B --> C )

Proof

Step Hyp Ref Expression
1 funcestrcsetc.e
 |-  E = ( ExtStrCat ` U )
2 funcestrcsetc.s
 |-  S = ( SetCat ` U )
3 funcestrcsetc.b
 |-  B = ( Base ` E )
4 funcestrcsetc.c
 |-  C = ( Base ` S )
5 funcestrcsetc.u
 |-  ( ph -> U e. WUni )
6 funcestrcsetc.f
 |-  ( ph -> F = ( x e. B |-> ( Base ` x ) ) )
7 1 3 5 estrcbasbas
 |-  ( ( 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 6 12 fmpt3d
 |-  ( ph -> F : B --> C )