Theorem fveu

Description: The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017)

	Ref	Expression
Assertion	fveu	⊢ ( ∃! 𝑥 𝐴 𝐹 𝑥 → ( 𝐹 ‘ 𝐴 ) = ∪ { 𝑥 ∣ 𝐴 𝐹 𝑥 } )

Proof

Step	Hyp	Ref	Expression
1		df-fv	⊢ ( 𝐹 ‘ 𝐴 ) = ( ℩ 𝑥 𝐴 𝐹 𝑥 )
2		iotauni	⊢ ( ∃! 𝑥 𝐴 𝐹 𝑥 → ( ℩ 𝑥 𝐴 𝐹 𝑥 ) = ∪ { 𝑥 ∣ 𝐴 𝐹 𝑥 } )
3	1 2	eqtrid	⊢ ( ∃! 𝑥 𝐴 𝐹 𝑥 → ( 𝐹 ‘ 𝐴 ) = ∪ { 𝑥 ∣ 𝐴 𝐹 𝑥 } )