Theorem preimafvsspwdm

Description: The class P of all preimages of function values is a subset of the power set of the domain of the function. (Contributed by AV, 5-Mar-2024)

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

Proof

Step	Hyp	Ref	Expression
1		setpreimafvex.p	⊢ 𝑃 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) }
2	1	elsetpreimafvssdm	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑠 ∈ 𝑃 ) → 𝑠 ⊆ 𝐴 )
3	2	ralrimiva	⊢ ( 𝐹 Fn 𝐴 → ∀ 𝑠 ∈ 𝑃 𝑠 ⊆ 𝐴 )
4		pwssb	⊢ ( 𝑃 ⊆ 𝒫 𝐴 ↔ ∀ 𝑠 ∈ 𝑃 𝑠 ⊆ 𝐴 )
5	3 4	sylibr	⊢ ( 𝐹 Fn 𝐴 → 𝑃 ⊆ 𝒫 𝐴 )