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Theorem elfv 6904
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 6901 . . 3 (𝐹𝐵) = {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)}
21eleq2i 2833 . 2 (𝐴 ∈ (𝐹𝐵) ↔ 𝐴 {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)})
3 eluniab 4921 . 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 206  wa 395  wal 1538  wex 1779  wcel 2108  {cab 2714   cuni 4907   class class class wbr 5143  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-sn 4627  df-uni 4908  df-iota 6514  df-fv 6569
This theorem is referenced by:  fv3  6924
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