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Theorem elfv 6880
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 6877 . . 3 (𝐹𝐵) = {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)}
21eleq2i 2817 . 2 (𝐴 ∈ (𝐹𝐵) ↔ 𝐴 {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)})
3 eluniab 4914 . 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 1531  wex 1773  wcel 2098  {cab 2701   cuni 4900   class class class wbr 5139  cfv 6534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3948  df-ss 3958  df-sn 4622  df-uni 4901  df-iota 6486  df-fv 6542
This theorem is referenced by:  fv3  6900
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