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Theorem elsetpreimafvb 47376
Description: The characterization of an element of the class 𝑃 of all preimages of function values. (Contributed by AV, 10-Mar-2024.)
Hypothesis
Ref Expression
setpreimafvex.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
Assertion
Ref Expression
elsetpreimafvb (𝑆𝑉 → (𝑆𝑃 ↔ ∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)})))
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝑥,𝑆,𝑧
Allowed substitution hints:   𝑃(𝑥,𝑧)   𝑉(𝑥,𝑧)

Proof of Theorem elsetpreimafvb
StepHypRef Expression
1 setpreimafvex.p . . 3 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
21eleq2i 2832 . 2 (𝑆𝑃𝑆 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})})
3 eqeq1 2740 . . . 4 (𝑧 = 𝑆 → (𝑧 = (𝐹 “ {(𝐹𝑥)}) ↔ 𝑆 = (𝐹 “ {(𝐹𝑥)})))
43rexbidv 3178 . . 3 (𝑧 = 𝑆 → (∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)}) ↔ ∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)})))
54elabg 3675 . 2 (𝑆𝑉 → (𝑆 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})} ↔ ∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)})))
62, 5bitrid 283 1 (𝑆𝑉 → (𝑆𝑃 ↔ ∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  wcel 2107  {cab 2713  wrex 3069  {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
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rex 3070
This theorem is referenced by:  elsetpreimafv  47377  preimafvelsetpreimafv  47380  0nelsetpreimafv  47382
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