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Theorem funmo 5287
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 5286 . . . . . 6  |-  ( Fun 
F  <->  ( Rel  F  /\  A. x E* y  x F y ) )
21simplbi 446 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
3 brrelex 4743 . . . . . 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 1621 . 2  |-  ( Fun 
F  ->  A. y
( A F y  ->  ( A  e. 
_V  /\  A F
y ) ) )
8 breq1 4042 . . . . . . 7  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
98mobidv 2191 . . . . . 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 1806 . . . . 5  |-  ( Fun 
F  ->  E* y  x F y )
1310, 12vtoclg 2856 . . . 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 2214 . . 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 2202 . 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 1530    = wceq 1632    e. wcel 1696   E*wmo 2157   _Vcvv 2801   class class class wbr 4039   Rel wrel 4710   Fun wfun 5265
This theorem is referenced by:  funeu  5294  funco  5308  imadif  5343  fneu  5364  dff3  5689  shftfn  11584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-fun 5273
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