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Theorem elfv 6889
Description: Membership in a function value. (Contributed by NM, 30-Apr-2004.)
Assertion
Ref Expression
elfv (𝐴 ∈ (𝐹𝐵) ↔ ∃𝑥(𝐴𝑥 ∧ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem elfv
StepHypRef Expression
1 fv2 6886 . . 3 (𝐹𝐵) = {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)}
21eleq2i 2821 . 2 (𝐴 ∈ (𝐹𝐵) ↔ 𝐴 {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)})
3 eluniab 4917 . 2 (𝐴 {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)} ↔ ∃𝑥(𝐴𝑥 ∧ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)))
42, 3bitri 275 1 (𝐴 ∈ (𝐹𝐵) ↔ ∃𝑥(𝐴𝑥 ∧ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wal 1532  wex 1774  wcel 2099  {cab 2705   cuni 4903   class class class wbr 5142  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-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3472  df-in 3952  df-ss 3962  df-sn 4625  df-uni 4904  df-iota 6494  df-fv 6550
This theorem is referenced by:  fv3  6909
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