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Theorem elfv 6845
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 6842 . . 3 (𝐹𝐵) = {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)}
21eleq2i 2830 . 2 (𝐴 ∈ (𝐹𝐵) ↔ 𝐴 {𝑥 ∣ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)})
3 eluniab 4885 . 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 397  wal 1540  wex 1782  wcel 2107  {cab 2714   cuni 4870   class class class wbr 5110  cfv 6501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3450  df-in 3922  df-ss 3932  df-sn 4592  df-uni 4871  df-iota 6453  df-fv 6509
This theorem is referenced by:  fv3  6865
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