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Metamath Proof Explorer

Theorem funcestrcsetclem1

Description: Lemma 1 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 funcestrcsetclem1 
|- ( ( ph /\ X e. B ) -> ( F ` X ) = ( Base ` X ) )

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 6 adantr 
 |-  ( ( ph /\ X e. B ) -> F = ( x e. B |-> ( Base ` x ) ) )
8 fveq2 
 |-  ( x = X -> ( Base ` x ) = ( Base ` X ) )
9 8 adantl 
 |-  ( ( ( ph /\ X e. B ) /\ x = X ) -> ( Base ` x ) = ( Base ` X ) )
10 simpr 
 |-  ( ( ph /\ X e. B ) -> X e. B )
11 fvexd 
 |-  ( ( ph /\ X e. B ) -> ( Base ` X ) e. _V )
12 7 9 10 11 fvmptd 
 |-  ( ( ph /\ X e. B ) -> ( F ` X ) = ( Base ` X ) )