Metamath Proof Explorer
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 |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
|
|
fnimasnd.2 |
⊢ ( 𝜑 → 𝑆 ∈ 𝐴 ) |
|
Assertion |
fnimasnd |
⊢ ( 𝜑 → ( 𝐹 “ { 𝑆 } ) = { ( 𝐹 ‘ 𝑆 ) } ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
fnimasnd.1 |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
2 |
|
fnimasnd.2 |
⊢ ( 𝜑 → 𝑆 ∈ 𝐴 ) |
3 |
|
fnsnfv |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝐴 ) → { ( 𝐹 ‘ 𝑆 ) } = ( 𝐹 “ { 𝑆 } ) ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → { ( 𝐹 ‘ 𝑆 ) } = ( 𝐹 “ { 𝑆 } ) ) |
5 |
4
|
eqcomd |
⊢ ( 𝜑 → ( 𝐹 “ { 𝑆 } ) = { ( 𝐹 ‘ 𝑆 ) } ) |