Theorem tz6.12-2-afv2

Description: Function value when F is (locally) not a function. Theorem 6.12(2) of TakeutiZaring p. 27, analogous to tz6.12-2 . (Contributed by AV, 5-Sep-2022)

	Ref	Expression
Assertion	tz6.12-2-afv2	⊢ ( ¬ ∃! 𝑥 𝐴 𝐹 𝑥 → ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		dfdfat2	⊢ ( 𝐹 defAt 𝐴 ↔ ( 𝐴 ∈ dom 𝐹 ∧ ∃! 𝑥 𝐴 𝐹 𝑥 ) )
2	1	simprbi	⊢ ( 𝐹 defAt 𝐴 → ∃! 𝑥 𝐴 𝐹 𝑥 )
3		ndfatafv2nrn	⊢ ( ¬ 𝐹 defAt 𝐴 → ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 )
4	2 3	nsyl5	⊢ ( ¬ ∃! 𝑥 𝐴 𝐹 𝑥 → ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 )