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Theorem fneu2 5550
Description: There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)
Assertion
Ref Expression
fneu2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! y <. B , 
y >.  e.  F )
Distinct variable groups:    y, F    y, B
Allowed substitution hint:    A( y)

Proof of Theorem fneu2
StepHypRef Expression
1 fneu 5549 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! y  B F y )
2 df-br 4213 . . 3  |-  ( B F y  <->  <. B , 
y >.  e.  F )
32eubii 2290 . 2  |-  ( E! y  B F y  <-> 
E! y <. B , 
y >.  e.  F )
41, 3sylib 189 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! y <. B , 
y >.  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   E!weu 2281   <.cop 3817   class class class wbr 4212    Fn wfn 5449
This theorem is referenced by:  feu  5619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-fun 5456  df-fn 5457
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