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Theorem funeu2 5613
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 4448 . 2  |-  ( A F B  <->  <. A ,  B >.  e.  F )
2 funeu 5612 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  E! y  A F y )
3 df-br 4448 . . . 4  |-  ( A F y  <->  <. A , 
y >.  e.  F )
43eubii 2300 . . 3  |-  ( E! y  A F y  <-> 
E! y <. A , 
y >.  e.  F )
52, 4sylib 196 . 2  |-  ( ( Fun  F  /\  A F B )  ->  E! y <. A ,  y
>.  e.  F )
61, 5sylan2br 476 1  |-  ( ( Fun  F  /\  <. A ,  B >.  e.  F
)  ->  E! y <. A ,  y >.  e.  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   E!weu 2275   <.cop 4033   class class class wbr 4447   Fun wfun 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-fun 5590
This theorem is referenced by:  funssres  5628
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