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Metamath Proof Explorer

Theorem wrd2pr2op

Description: A word of length two represented as unordered pair of ordered pairs. (Contributed by AV, 20-Oct-2018) (Proof shortened by AV, 26-Jan-2021)

Ref Expression
Assertion wrd2pr2op 
|- ( ( W e. Word V /\ ( # ` W ) = 2 ) -> W = { <. 0 , ( W ` 0 ) >. , <. 1 , ( W ` 1 ) >. } )

Proof

Step Hyp Ref Expression
1 wrdfn 
 |-  ( W e. Word V -> W Fn ( 0 ..^ ( # ` W ) ) )
2 1 adantr 
 |-  ( ( W e. Word V /\ ( # ` W ) = 2 ) -> W Fn ( 0 ..^ ( # ` W ) ) )
3 oveq2 
 |-  ( ( # ` W ) = 2 -> ( 0 ..^ ( # ` W ) ) = ( 0 ..^ 2 ) )
4 fzo0to2pr 
 |-  ( 0 ..^ 2 ) = { 0 , 1 }
5 3 4 eqtr2di 
 |-  ( ( # ` W ) = 2 -> { 0 , 1 } = ( 0 ..^ ( # ` W ) ) )
6 5 adantl 
 |-  ( ( W e. Word V /\ ( # ` W ) = 2 ) -> { 0 , 1 } = ( 0 ..^ ( # ` W ) ) )
7 6 fneq2d 
 |-  ( ( W e. Word V /\ ( # ` W ) = 2 ) -> ( W Fn { 0 , 1 } <-> W Fn ( 0 ..^ ( # ` W ) ) ) )
8 2 7 mpbird 
 |-  ( ( W e. Word V /\ ( # ` W ) = 2 ) -> W Fn { 0 , 1 } )
9 c0ex 
 |-  0 e. _V
10 1ex 
 |-  1 e. _V
11 9 10 fnprb 
 |-  ( W Fn { 0 , 1 } <-> W = { <. 0 , ( W ` 0 ) >. , <. 1 , ( W ` 1 ) >. } )
12 8 11 sylib 
 |-  ( ( W e. Word V /\ ( # ` W ) = 2 ) -> W = { <. 0 , ( W ` 0 ) >. , <. 1 , ( W ` 1 ) >. } )