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Metamath Proof Explorer

Theorem s1dm

Description: The domain of a singleton word is a singleton. (Contributed by AV, 9-Jan-2020)

Ref Expression
Assertion s1dm 
|- dom <" A "> = { 0 }

Proof

Step Hyp Ref Expression
1 s1cli 
 |-  <" A "> e. Word _V
2 wrdf 
 |-  ( <" A "> e. Word _V -> <" A "> : ( 0 ..^ ( # ` <" A "> ) ) --> _V )
3 1 2 ax-mp 
 |-  <" A "> : ( 0 ..^ ( # ` <" A "> ) ) --> _V
4 s1len 
 |-  ( # ` <" A "> ) = 1
5 oveq2 
 |-  ( ( # ` <" A "> ) = 1 -> ( 0 ..^ ( # ` <" A "> ) ) = ( 0 ..^ 1 ) )
6 fzo01 
 |-  ( 0 ..^ 1 ) = { 0 }
7 5 6 eqtrdi 
 |-  ( ( # ` <" A "> ) = 1 -> ( 0 ..^ ( # ` <" A "> ) ) = { 0 } )
8 4 7 ax-mp 
 |-  ( 0 ..^ ( # ` <" A "> ) ) = { 0 }
9 8 eqcomi 
 |-  { 0 } = ( 0 ..^ ( # ` <" A "> ) )
10 9 feq2i 
 |-  ( <" A "> : { 0 } --> _V <-> <" A "> : ( 0 ..^ ( # ` <" A "> ) ) --> _V )
11 3 10 mpbir 
 |-  <" A "> : { 0 } --> _V
12 11 fdmi 
 |-  dom <" A "> = { 0 }