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 ) >. } )