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Metamath Proof Explorer

Theorem wrdco

Description: Mapping a word by a function. (Contributed by Stefan O'Rear, 27-Aug-2015)

Ref Expression
Assertion wrdco 
|- ( ( W e. Word A /\ F : A --> B ) -> ( F o. W ) e. Word B )

Proof

Step Hyp Ref Expression
1 simpr 
 |-  ( ( W e. Word A /\ F : A --> B ) -> F : A --> B )
2 wrdf 
 |-  ( W e. Word A -> W : ( 0 ..^ ( # ` W ) ) --> A )
3 2 adantr 
 |-  ( ( W e. Word A /\ F : A --> B ) -> W : ( 0 ..^ ( # ` W ) ) --> A )
4 fco 
 |-  ( ( F : A --> B /\ W : ( 0 ..^ ( # ` W ) ) --> A ) -> ( F o. W ) : ( 0 ..^ ( # ` W ) ) --> B )
5 1 3 4 syl2anc 
 |-  ( ( W e. Word A /\ F : A --> B ) -> ( F o. W ) : ( 0 ..^ ( # ` W ) ) --> B )
6 iswrdi 
 |-  ( ( F o. W ) : ( 0 ..^ ( # ` W ) ) --> B -> ( F o. W ) e. Word B )
7 5 6 syl 
 |-  ( ( W e. Word A /\ F : A --> B ) -> ( F o. W ) e. Word B )