Metamath Proof Explorer


Theorem funcestrcsetclem2

Description: Lemma 2 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 funcestrcsetclem2
|- ( ( ph /\ X e. B ) -> ( F ` X ) e. U )

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 2 3 4 5 6 funcestrcsetclem1
 |-  ( ( ph /\ X e. B ) -> ( F ` X ) = ( Base ` X ) )
8 1 3 5 estrcbasbas
 |-  ( ( ph /\ X e. B ) -> ( Base ` X ) e. U )
9 7 8 eqeltrd
 |-  ( ( ph /\ X e. B ) -> ( F ` X ) e. U )