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Metamath Proof Explorer

Theorem wrdval

Description: Value of the set of words over a set. (Contributed by Stefan O'Rear, 10-Aug-2015) (Revised by Mario Carneiro, 26-Feb-2016)

Ref Expression
Assertion wrdval 
|- ( S e. V -> Word S = U_ l e. NN0 ( S ^m ( 0 ..^ l ) ) )

Proof

Step Hyp Ref Expression
1 df-word 
 |-  Word S = { w | E. l e. NN0 w : ( 0 ..^ l ) --> S }
2 eliun 
 |-  ( w e. U_ l e. NN0 ( S ^m ( 0 ..^ l ) ) <-> E. l e. NN0 w e. ( S ^m ( 0 ..^ l ) ) )
3 ovex 
 |-  ( 0 ..^ l ) e. _V
4 elmapg 
 |-  ( ( S e. V /\ ( 0 ..^ l ) e. _V ) -> ( w e. ( S ^m ( 0 ..^ l ) ) <-> w : ( 0 ..^ l ) --> S ) )
5 3 4 mpan2 
 |-  ( S e. V -> ( w e. ( S ^m ( 0 ..^ l ) ) <-> w : ( 0 ..^ l ) --> S ) )
6 5 rexbidv 
 |-  ( S e. V -> ( E. l e. NN0 w e. ( S ^m ( 0 ..^ l ) ) <-> E. l e. NN0 w : ( 0 ..^ l ) --> S ) )
7 2 6 bitrid 
 |-  ( S e. V -> ( w e. U_ l e. NN0 ( S ^m ( 0 ..^ l ) ) <-> E. l e. NN0 w : ( 0 ..^ l ) --> S ) )
8 7 eqabdv 
 |-  ( S e. V -> U_ l e. NN0 ( S ^m ( 0 ..^ l ) ) = { w | E. l e. NN0 w : ( 0 ..^ l ) --> S } )
9 1 8 eqtr4id 
 |-  ( S e. V -> Word S = U_ l e. NN0 ( S ^m ( 0 ..^ l ) ) )