Theorem elsetpreimafv

Description: An element of the class P of all preimages of function values. (Contributed by AV, 8-Mar-2024)

	Ref	Expression
Hypothesis	setpreimafvex.p	⊢ 𝑃 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) }
Assertion	elsetpreimafv	⊢ ( 𝑆 ∈ 𝑃 → ∃ 𝑥 ∈ 𝐴 𝑆 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) )

Proof

Step	Hyp	Ref	Expression
1		setpreimafvex.p	⊢ 𝑃 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) }
2	1	elsetpreimafvb	⊢ ( 𝑆 ∈ 𝑃 → ( 𝑆 ∈ 𝑃 ↔ ∃ 𝑥 ∈ 𝐴 𝑆 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) ) )
3	2	ibi	⊢ ( 𝑆 ∈ 𝑃 → ∃ 𝑥 ∈ 𝐴 𝑆 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) )