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

Theorem funmo 5403
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 5402 . . . . . 6  |-  ( Fun 
F  <->  ( Rel  F  /\  A. x E* y  x F y ) )
21simplbi 447 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
3 brrelex 4849 . . . . . 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 1638 . 2  |-  ( Fun 
F  ->  A. y
( A F y  ->  ( A  e. 
_V  /\  A F
y ) ) )
8 breq1 4149 . . . . . . 7  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
98mobidv 2266 . . . . . 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 1766 . . . . 5  |-  ( Fun 
F  ->  E* y  x F y )
1310, 12vtoclg 2947 . . . 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 2289 . . 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 2277 . 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 1546    = wceq 1649    e. wcel 1717   E*wmo 2232   _Vcvv 2892   class class class wbr 4146   Rel wrel 4816   Fun wfun 5381
This theorem is referenced by:  funeu  5410  funco  5424  imadif  5461  fneu  5482  dff3  5814  shftfn  11808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-fun 5389
  Copyright terms: Public domain W3C validator