Theorem s2dm

Description: The domain of a doubleton word is an unordered pair. (Contributed by AV, 9-Jan-2020)

	Ref	Expression
Assertion	s2dm	|- dom <" A B "> = { 0 , 1 }

Proof

Step	Hyp	Ref	Expression
1		s2cli	|- <" A B "> e. Word _V
2		wrdf	|- ( <" A B "> e. Word _V -> <" A B "> : ( 0 ..^ ( # ` <" A B "> ) ) --> _V )
3	1 2	ax-mp	|- <" A B "> : ( 0 ..^ ( # ` <" A B "> ) ) --> _V
4		s2len	|- ( # ` <" A B "> ) = 2
5		oveq2	|- ( ( # ` <" A B "> ) = 2 -> ( 0 ..^ ( # ` <" A B "> ) ) = ( 0 ..^ 2 ) )
6		fzo0to2pr	|- ( 0 ..^ 2 ) = { 0 , 1 }
7	5 6	eqtrdi	|- ( ( # ` <" A B "> ) = 2 -> ( 0 ..^ ( # ` <" A B "> ) ) = { 0 , 1 } )
8	4 7	ax-mp	|- ( 0 ..^ ( # ` <" A B "> ) ) = { 0 , 1 }
9	8	feq2i	|- ( <" A B "> : ( 0 ..^ ( # ` <" A B "> ) ) --> _V <-> <" A B "> : { 0 , 1 } --> _V )
10	3 9	mpbi	|- <" A B "> : { 0 , 1 } --> _V
11	10	fdmi	|- dom <" A B "> = { 0 , 1 }