Theorem afv2fvn0fveq

Description: If the function's value at an argument is not the empty set, it equals the alternate function value at this argument. (Contributed by AV, 3-Sep-2022)

	Ref	Expression
Assertion	afv2fvn0fveq	⊢ ( ( 𝐹 ‘ 𝐴 ) ≠ ∅ → ( 𝐹 '''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		fvfundmfvn0	⊢ ( ( 𝐹 ‘ 𝐴 ) ≠ ∅ → ( 𝐴 ∈ dom 𝐹 ∧ Fun ( 𝐹 ↾ { 𝐴 } ) ) )
2		df-dfat	⊢ ( 𝐹 defAt 𝐴 ↔ ( 𝐴 ∈ dom 𝐹 ∧ Fun ( 𝐹 ↾ { 𝐴 } ) ) )
3	1 2	sylibr	⊢ ( ( 𝐹 ‘ 𝐴 ) ≠ ∅ → 𝐹 defAt 𝐴 )
4		dfatafv2eqfv	⊢ ( 𝐹 defAt 𝐴 → ( 𝐹 '''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )
5	3 4	syl	⊢ ( ( 𝐹 ‘ 𝐴 ) ≠ ∅ → ( 𝐹 '''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )