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Theorem elfv 5326
 Description: Membership in a function value. (Contributed by set.mm contributors, 30-Apr-2004.)
Assertion
Ref Expression
elfv (A (FB) ↔ x(A x 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 5324 . . 3 (FB) = {x y(BFyy = x)}
21eleq2i 2417 . 2 (A (FB) ↔ A {x y(BFyy = x)})
3 eluniab 3903 . 2 (A {x y(BFyy = x)} ↔ x(A x y(BFyy = x)))
42, 3bitri 240 1 (A (FB) ↔ x(A x y(BFyy = x)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   ∈ wcel 1710  {cab 2339  ∪cuni 3891   class class class wbr 4639   ‘cfv 4781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-sn 3741  df-uni 3892  df-iota 4339  df-fv 4795 This theorem is referenced by:  fv3  5341
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