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Theorem elsetpreimafv 46725
Description: An element of the class 𝑃 of all preimages of function values. (Contributed by AV, 8-Mar-2024.)
Hypothesis
Ref Expression
setpreimafvex.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
Assertion
Ref Expression
elsetpreimafv (𝑆𝑃 → ∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)}))
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝑥,𝑆,𝑧
Allowed substitution hints:   𝑃(𝑥,𝑧)

Proof of Theorem elsetpreimafv
StepHypRef Expression
1 setpreimafvex.p . . 3 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
21elsetpreimafvb 46724 . 2 (𝑆𝑃 → (𝑆𝑃 ↔ ∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)})))
32ibi 267 1 (𝑆𝑃 → ∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  {cab 2705  wrex 3067  {csn 4629  ccnv 5677  cima 5681  cfv 6548
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 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rex 3068
This theorem is referenced by:  elsetpreimafvssdm  46726  fvelsetpreimafv  46727  elsetpreimafvbi  46731  imasetpreimafvbijlemfo  46745
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