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Theorem funmo 5462
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 5461 . . . . . 6  |-  ( Fun 
F  <->  ( Rel  F  /\  A. x E* y  x F y ) )
21simplbi 447 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
3 brrelex 4908 . . . . . 6  |-  ( ( Rel  F  /\  A F y )  ->  A  e.  _V )
43ex 424 . . . . 5  |-  ( Rel 
F  ->  ( A F y  ->  A  e.  _V ) )
52, 4syl 16 . . . 4  |-  ( Fun 
F  ->  ( A F y  ->  A  e.  _V ) )
65ancrd 538 . . 3  |-  ( Fun 
F  ->  ( A F y  ->  ( A  e.  _V  /\  A F y ) ) )
76alrimiv 1641 . 2  |-  ( Fun 
F  ->  A. y
( A F y  ->  ( A  e. 
_V  /\  A F
y ) ) )
8 breq1 4207 . . . . . . 7  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
98mobidv 2315 . . . . . 6  |-  ( x  =  A  ->  ( E* y  x F
y  <->  E* y  A F y ) )
109imbi2d 308 . . . . 5  |-  ( x  =  A  ->  (
( Fun  F  ->  E* y  x F y )  <->  ( Fun  F  ->  E* y  A F y ) ) )
111simprbi 451 . . . . . 6  |-  ( Fun 
F  ->  A. x E* y  x F
y )
121119.21bi 1774 . . . . 5  |-  ( Fun 
F  ->  E* y  x F y )
1310, 12vtoclg 3003 . . . 4  |-  ( A  e.  _V  ->  ( Fun  F  ->  E* y  A F y ) )
1413com12 29 . . 3  |-  ( Fun 
F  ->  ( A  e.  _V  ->  E* y  A F y ) )
15 moanimv 2338 . . 3  |-  ( E* y ( A  e. 
_V  /\  A F
y )  <->  ( A  e.  _V  ->  E* y  A F y ) )
1614, 15sylibr 204 . 2  |-  ( Fun 
F  ->  E* y
( A  e.  _V  /\  A F y ) )
17 moim 2326 . 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 58 1  |-  ( Fun 
F  ->  E* y  A F y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   E*wmo 2281   _Vcvv 2948   class class class wbr 4204   Rel wrel 4875   Fun wfun 5440
This theorem is referenced by:  funeu  5469  funco  5483  imadif  5520  fneu  5541  dff3  5874  shftfn  11880
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-fun 5448
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