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Theorem elfv 6227
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 6224 . . 3 (𝐹𝐵) = {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)}
21eleq2i 2722 . 2 (𝐴 ∈ (𝐹𝐵) ↔ 𝐴 {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)})
3 eluniab 4479 . 2 (𝐴 {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)} ↔ ∃𝑥(𝐴𝑥 ∧ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)))
42, 3bitri 264 1 (𝐴 ∈ (𝐹𝐵) ↔ ∃𝑥(𝐴𝑥 ∧ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wal 1521  wex 1744  wcel 2030  {cab 2637   cuni 4468   class class class wbr 4685  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-v 3233  df-sn 4211  df-uni 4469  df-iota 5889  df-fv 5934
This theorem is referenced by:  fv3  6244
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