Theorem elsetpreimafvb

Description: The characterization of an element of the class P of all preimages of function values. (Contributed by AV, 10-Mar-2024)

	Ref	Expression
Hypothesis	setpreimafvex.p	⊢ 𝑃 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) }
Assertion	elsetpreimafvb	⊢ ( 𝑆 ∈ 𝑉 → ( 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	eleq2i	⊢ ( 𝑆 ∈ 𝑃 ↔ 𝑆 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) } )
3		eqeq1	⊢ ( 𝑧 = 𝑆 → ( 𝑧 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) ↔ 𝑆 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) ) )
4	3	rexbidv	⊢ ( 𝑧 = 𝑆 → ( ∃ 𝑥 ∈ 𝐴 𝑧 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) ↔ ∃ 𝑥 ∈ 𝐴 𝑆 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) ) )
5	4	elabg	⊢ ( 𝑆 ∈ 𝑉 → ( 𝑆 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) } ↔ ∃ 𝑥 ∈ 𝐴 𝑆 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) ) )
6	2 5	syl5bb	⊢ ( 𝑆 ∈ 𝑉 → ( 𝑆 ∈ 𝑃 ↔ ∃ 𝑥 ∈ 𝐴 𝑆 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) ) )