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Theorem funeu2 5470
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 4205 . 2  |-  ( A F B  <->  <. A ,  B >.  e.  F )
2 funeu 5469 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  E! y  A F y )
3 df-br 4205 . . . 4  |-  ( A F y  <->  <. A , 
y >.  e.  F )
43eubii 2289 . . 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 1725   E!weu 2280   <.cop 3809   class class class wbr 4204   Fun wfun 5440
This theorem is referenced by:  funssres  5485
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-fun 5448
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