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Theorem elfv 5097
 Description: Membership in a function value. (Contributed by NM, 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 5094 . . 3 (𝐹B) = {xy(B𝐹yy = x)}
21eleq2i 2082 . 2 (A (𝐹B) ↔ A {xy(B𝐹yy = x)})
3 eluniab 3562 . 2 (A {xy(B𝐹yy = x)} ↔ x(A x y(B𝐹yy = x)))
42, 3bitri 173 1 (A (𝐹B) ↔ x(A x y(B𝐹yy = x)))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∀wal 1224  ∃wex 1358   ∈ wcel 1370  {cab 2004  ∪ cuni 3550   class class class wbr 3734  ‘cfv 4825 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-v 2533  df-sn 3352  df-uni 3551  df-iota 4790  df-fv 4833 This theorem is referenced by:  fv3  5118  relelfvdm  5126
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