Metamath Proof Explorer

Theorem wrd3tpop

Description: A word of length three represented as triple of ordered pairs. (Contributed by AV, 26-Jan-2021)

Ref Expression
Assertion wrd3tpop 
|- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> W = { <. 0 , ( W ` 0 ) >. , <. 1 , ( W ` 1 ) >. , <. 2 , ( W ` 2 ) >. } )

Proof

Step Hyp Ref Expression
1 wrdfn 
 |-  ( W e. Word V -> W Fn ( 0 ..^ ( # ` W ) ) )
2 1 adantr 
 |-  ( ( W e. Word V /\ ( # ` W ) = 3 ) -> W Fn ( 0 ..^ ( # ` W ) ) )
3 oveq2 
 |-  ( ( # ` W ) = 3 -> ( 0 ..^ ( # ` W ) ) = ( 0 ..^ 3 ) )
4 fzo0to3tp 
 |-  ( 0 ..^ 3 ) = { 0 , 1 , 2 }
5 3 4 eqtr2di 
 |-  ( ( # ` W ) = 3 -> { 0 , 1 , 2 } = ( 0 ..^ ( # ` W ) ) )
6 5 adantl 
 |-  ( ( W e. Word V /\ ( # ` W ) = 3 ) -> { 0 , 1 , 2 } = ( 0 ..^ ( # ` W ) ) )
7 6 fneq2d 
 |-  ( ( W e. Word V /\ ( # ` W ) = 3 ) -> ( W Fn { 0 , 1 , 2 } <-> W Fn ( 0 ..^ ( # ` W ) ) ) )
8 2 7 mpbird 
 |-  ( ( W e. Word V /\ ( # ` W ) = 3 ) -> W Fn { 0 , 1 , 2 } )
9 c0ex 
 |-  0 e. _V
10 1ex 
 |-  1 e. _V
11 2ex 
 |-  2 e. _V
12 9 10 11 fntpb 
 |-  ( W Fn { 0 , 1 , 2 } <-> W = { <. 0 , ( W ` 0 ) >. , <. 1 , ( W ` 1 ) >. , <. 2 , ( W ` 2 ) >. } )
13 8 12 sylib 
 |-  ( ( W e. Word V /\ ( # ` W ) = 3 ) -> W = { <. 0 , ( W ` 0 ) >. , <. 1 , ( W ` 1 ) >. , <. 2 , ( W ` 2 ) >. } )