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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 ∣ ∃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 7986 . 2 (𝐴𝑉 → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ∈ V)
31, 2eqeltrid 2844 1 (𝐴𝑉𝑃 ∈ V)
Colors of variables: wff setvar class
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
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