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Metamath Proof Explorer

Theorem dprdfcl

Description: A finitely supported function in S has its X -th element in S ( X ) . (Contributed by Mario Carneiro, 25-Apr-2016) (Revised by AV, 11-Jul-2019)

Ref Expression
Hypotheses dprdff.w 
|- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
dprdff.1 
|- ( ph -> G dom DProd S )
dprdff.2 
|- ( ph -> dom S = I )
dprdff.3 
|- ( ph -> F e. W )
Assertion dprdfcl 
|- ( ( ph /\ X e. I ) -> ( F ` X ) e. ( S ` X ) )

Proof

Step Hyp Ref Expression
1 dprdff.w 
 |-  W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
2 dprdff.1 
 |-  ( ph -> G dom DProd S )
3 dprdff.2 
 |-  ( ph -> dom S = I )
4 dprdff.3 
 |-  ( ph -> F e. W )
5 1 2 3 dprdw 
 |-  ( ph -> ( F e. W <-> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) /\ F finSupp .0. ) ) )
6 4 5 mpbid 
 |-  ( ph -> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) /\ F finSupp .0. ) )
7 6 simp2d 
 |-  ( ph -> A. x e. I ( F ` x ) e. ( S ` x ) )
8 fveq2 
 |-  ( x = X -> ( F ` x ) = ( F ` X ) )
9 fveq2 
 |-  ( x = X -> ( S ` x ) = ( S ` X ) )
10 8 9 eleq12d 
 |-  ( x = X -> ( ( F ` x ) e. ( S ` x ) <-> ( F ` X ) e. ( S ` X ) ) )
11 10 rspccva 
 |-  ( ( A. x e. I ( F ` x ) e. ( S ` x ) /\ X e. I ) -> ( F ` X ) e. ( S ` X ) )
12 7 11 sylan 
 |-  ( ( ph /\ X e. I ) -> ( F ` X ) e. ( S ` X ) )