Theorem setpreimafvex 47872

Description: The class 𝑃 of all preimages of function values is a set. (Contributed by AV, 10-Mar-2024.)

Hypothesis

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

Assertion

Ref	Expression
setpreimafvex	⊢ (𝐴 ∈ 𝑉 → 𝑃 ∈ V)

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧

Allowed substitution hints:   𝑃(𝑥,𝑧)   𝑉(𝑥,𝑧)

Proof of Theorem setpreimafvex

Step	Hyp	Ref	Expression
1		setpreimafvex.p	. 2 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
2		abrexexg 7907	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ∈ V)
3	1, 2	eqeltrid 2845	1 ⊢ (𝐴 ∈ 𝑉 → 𝑃 ∈ V)

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121  {cab 2719  ∃wrex 3065  Vcvv 3433  {csn 4558  ^◡ccnv 5620   “ cima 5624  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-rep 5202

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-mo 2545  df-clab 2720  df-cleq 2733  df-clel 2816  df-rex 3066  df-v 3435

This theorem is referenced by:  fundcmpsurbijinjpreimafv  47896