Theorem setpreimafvex 47257

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 8001	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ∈ V)
3	1, 2	eqeltrid 2848	1 ⊢ (𝐴 ∈ 𝑉 → 𝑃 ∈ V)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  {cab 2717  ∃wrex 3076  Vcvv 3488  {csn 4648  ^◡ccnv 5699   “ cima 5703  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-rep 5303

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-rex 3077  df-v 3490

This theorem is referenced by:  fundcmpsurbijinjpreimafv  47281