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Theorem funmo 5271
Description: A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)
Assertion
Ref Expression
funmo  |-  ( Fun 
F  ->  E* y  A F y )
Distinct variable groups:    y, A    y, F

Proof of Theorem funmo
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dffun6 5270 . . . . . 6  |-  ( Fun 
F  <->  ( Rel  F  /\  A. x E* y  x F y ) )
21simplbi 446 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
3 brrelex 4727 . . . . . 6  |-  ( ( Rel  F  /\  A F y )  ->  A  e.  _V )
43ex 423 . . . . 5  |-  ( Rel 
F  ->  ( A F y  ->  A  e.  _V ) )
52, 4syl 15 . . . 4  |-  ( Fun 
F  ->  ( A F y  ->  A  e.  _V ) )
65ancrd 537 . . 3  |-  ( Fun 
F  ->  ( A F y  ->  ( A  e.  _V  /\  A F y ) ) )
76alrimiv 1617 . 2  |-  ( Fun 
F  ->  A. y
( A F y  ->  ( A  e. 
_V  /\  A F
y ) ) )
8 breq1 4026 . . . . . . 7  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
98mobidv 2178 . . . . . 6  |-  ( x  =  A  ->  ( E* y  x F
y  <->  E* y  A F y ) )
109imbi2d 307 . . . . 5  |-  ( x  =  A  ->  (
( Fun  F  ->  E* y  x F y )  <->  ( Fun  F  ->  E* y  A F y ) ) )
111simprbi 450 . . . . . 6  |-  ( Fun 
F  ->  A. x E* y  x F
y )
121119.21bi 1794 . . . . 5  |-  ( Fun 
F  ->  E* y  x F y )
1310, 12vtoclg 2843 . . . 4  |-  ( A  e.  _V  ->  ( Fun  F  ->  E* y  A F y ) )
1413com12 27 . . 3  |-  ( Fun 
F  ->  ( A  e.  _V  ->  E* y  A F y ) )
15 moanimv 2201 . . 3  |-  ( E* y ( A  e. 
_V  /\  A F
y )  <->  ( A  e.  _V  ->  E* y  A F y ) )
1614, 15sylibr 203 . 2  |-  ( Fun 
F  ->  E* y
( A  e.  _V  /\  A F y ) )
17 moim 2189 . 2  |-  ( A. y ( A F y  ->  ( A  e.  _V  /\  A F y ) )  -> 
( E* y ( A  e.  _V  /\  A F y )  ->  E* y  A F
y ) )
187, 16, 17sylc 56 1  |-  ( Fun 
F  ->  E* y  A F y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   E*wmo 2144   _Vcvv 2788   class class class wbr 4023   Rel wrel 4694   Fun wfun 5249
This theorem is referenced by:  funeu  5278  funco  5292  imadif  5327  fneu  5348  dff3  5673  shftfn  11568
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-fun 5257
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