Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  setpreimafvex Structured version   Visualization version   GIF version

Theorem setpreimafvex 46646
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 7958 . 2 (𝐴𝑉 → {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ∈ V)
31, 2eqeltrid 2832 1 (𝐴𝑉𝑃 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  {cab 2704  wrex 3065  Vcvv 3469  {csn 4624  ccnv 5671  cima 5675  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-rep 5279
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-mo 2529  df-clab 2705  df-cleq 2719  df-clel 2805  df-rex 3066  df-v 3471
This theorem is referenced by:  fundcmpsurbijinjpreimafv  46670
  Copyright terms: Public domain W3C validator