Theorem dfatprc

Description: A function is not defined at a proper class. (Contributed by AV, 1-Sep-2022)

	Ref	Expression
Assertion	dfatprc	⊢ ( ¬ 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		prcnel	⊢ ( ¬ 𝐴 ∈ V → ¬ 𝐴 ∈ dom 𝐹 )
2	1	orcd	⊢ ( ¬ 𝐴 ∈ V → ( ¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun ( 𝐹 ↾ { 𝐴 } ) ) )
3		ianor	⊢ ( ¬ ( 𝐴 ∈ dom 𝐹 ∧ Fun ( 𝐹 ↾ { 𝐴 } ) ) ↔ ( ¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun ( 𝐹 ↾ { 𝐴 } ) ) )
4		df-dfat	⊢ ( 𝐹 defAt 𝐴 ↔ ( 𝐴 ∈ dom 𝐹 ∧ Fun ( 𝐹 ↾ { 𝐴 } ) ) )
5	3 4	xchnxbir	⊢ ( ¬ 𝐹 defAt 𝐴 ↔ ( ¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun ( 𝐹 ↾ { 𝐴 } ) ) )
6	2 5	sylibr	⊢ ( ¬ 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴 )