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Theorem setpreimafvex 46502
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 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
Assertion
Ref Expression
setpreimafvex (𝐴𝑉𝑃 ∈ V)
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧
Allowed substitution hints:   𝑃(𝑥,𝑧)   𝑉(𝑥,𝑧)

Proof of Theorem setpreimafvex
StepHypRef Expression
1 setpreimafvex.p . 2 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
2 abrexexg 7940 . 2 (𝐴𝑉 → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ∈ V)
31, 2eqeltrid 2829 1 (𝐴𝑉𝑃 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {cab 2701  wrex 3062  Vcvv 3466  {csn 4620  ccnv 5665  cima 5669  cfv 6533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-rep 5275
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-mo 2526  df-clab 2702  df-cleq 2716  df-clel 2802  df-rex 3063  df-v 3468
This theorem is referenced by:  fundcmpsurbijinjpreimafv  46526
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