Metamath Proof Explorer


Theorem fnimasnd

Description: The image of a function by a singleton whose element is in the domain of the function. (Contributed by Steven Nguyen, 7-Jun-2023)

Ref Expression
Hypotheses fnimasnd.1 ( 𝜑𝐹 Fn 𝐴 )
fnimasnd.2 ( 𝜑𝑆𝐴 )
Assertion fnimasnd ( 𝜑 → ( 𝐹 “ { 𝑆 } ) = { ( 𝐹𝑆 ) } )

Proof

Step Hyp Ref Expression
1 fnimasnd.1 ( 𝜑𝐹 Fn 𝐴 )
2 fnimasnd.2 ( 𝜑𝑆𝐴 )
3 fnsnfv ( ( 𝐹 Fn 𝐴𝑆𝐴 ) → { ( 𝐹𝑆 ) } = ( 𝐹 “ { 𝑆 } ) )
4 1 2 3 syl2anc ( 𝜑 → { ( 𝐹𝑆 ) } = ( 𝐹 “ { 𝑆 } ) )
5 4 eqcomd ( 𝜑 → ( 𝐹 “ { 𝑆 } ) = { ( 𝐹𝑆 ) } )