MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funeu Unicode version

Theorem funeu 5278
Description: There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
funeu  |-  ( ( Fun  F  /\  A F B )  ->  E! y  A F y )
Distinct variable groups:    y, A    y, F
Allowed substitution hint:    B( y)

Proof of Theorem funeu
StepHypRef Expression
1 funrel 5272 . . . 4  |-  ( Fun 
F  ->  Rel  F )
2 releldm 4911 . . . 4  |-  ( ( Rel  F  /\  A F B )  ->  A  e.  dom  F )
31, 2sylan 457 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  A  e.  dom  F )
4 eldmg 4874 . . . 4  |-  ( A  e.  dom  F  -> 
( A  e.  dom  F  <->  E. y  A F
y ) )
54ibi 232 . . 3  |-  ( A  e.  dom  F  ->  E. y  A F
y )
63, 5syl 15 . 2  |-  ( ( Fun  F  /\  A F B )  ->  E. y  A F y )
7 funmo 5271 . . . 4  |-  ( Fun 
F  ->  E* y  A F y )
87adantr 451 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  E* y  A F y )
9 df-mo 2148 . . 3  |-  ( E* y  A F y  <-> 
( E. y  A F y  ->  E! y  A F y ) )
108, 9sylib 188 . 2  |-  ( ( Fun  F  /\  A F B )  ->  ( E. y  A F
y  ->  E! y  A F y ) )
116, 10mpd 14 1  |-  ( ( Fun  F  /\  A F B )  ->  E! y  A F y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    e. wcel 1684   E!weu 2143   E*wmo 2144   class class class wbr 4023   dom cdm 4689   Rel wrel 4694   Fun wfun 5249
This theorem is referenced by:  funeu2  5279  funbrfv  5561
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257
  Copyright terms: Public domain W3C validator