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Theorem funeu2 5419
Description: There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)
Assertion
Ref Expression
funeu2  |-  ( ( Fun  F  /\  <. A ,  B >.  e.  F
)  ->  E! y <. A ,  y >.  e.  F )
Distinct variable groups:    y, A    y, F
Allowed substitution hint:    B( y)

Proof of Theorem funeu2
StepHypRef Expression
1 df-br 4155 . 2  |-  ( A F B  <->  <. A ,  B >.  e.  F )
2 funeu 5418 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  E! y  A F y )
3 df-br 4155 . . . 4  |-  ( A F y  <->  <. A , 
y >.  e.  F )
43eubii 2248 . . 3  |-  ( E! y  A F y  <-> 
E! y <. A , 
y >.  e.  F )
52, 4sylib 189 . 2  |-  ( ( Fun  F  /\  A F B )  ->  E! y <. A ,  y
>.  e.  F )
61, 5sylan2br 463 1  |-  ( ( Fun  F  /\  <. A ,  B >.  e.  F
)  ->  E! y <. A ,  y >.  e.  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   E!weu 2239   <.cop 3761   class class class wbr 4154   Fun wfun 5389
This theorem is referenced by:  funssres  5434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-fun 5397
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