Theorem fundmdfat

Description: A function is defined at any element of its domain. (Contributed by AV, 2-Sep-2022)

	Ref	Expression
Assertion	fundmdfat	⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → 𝐹 defAt 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		funres	⊢ ( Fun 𝐹 → Fun ( 𝐹 ↾ { 𝐴 } ) )
2	1	anim1ci	⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → ( 𝐴 ∈ dom 𝐹 ∧ Fun ( 𝐹 ↾ { 𝐴 } ) ) )
3		df-dfat	⊢ ( 𝐹 defAt 𝐴 ↔ ( 𝐴 ∈ dom 𝐹 ∧ Fun ( 𝐹 ↾ { 𝐴 } ) ) )
4	2 3	sylibr	⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → 𝐹 defAt 𝐴 )