Theorem setpreimafvex 47375

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 7986	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ∈ V)
3	1, 2	eqeltrid 2844	1 ⊢ (𝐴 ∈ 𝑉 → 𝑃 ∈ V)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  {cab 2713  ∃wrex 3069  Vcvv 3479  {csn 4625  ^◡ccnv 5683   “ cima 5687  ‘cfv 6560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-rep 5278

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-rex 3070  df-v 3481

This theorem is referenced by:  fundcmpsurbijinjpreimafv  47399