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

Theorem funmo 5605
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 5604 . . . . . 6  |-  ( Fun 
F  <->  ( Rel  F  /\  A. x E* y  x F y ) )
21simplbi 467 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
3 brrelex 4878 . . . . . 6  |-  ( ( Rel  F  /\  A F y )  ->  A  e.  _V )
43ex 441 . . . . 5  |-  ( Rel 
F  ->  ( A F y  ->  A  e.  _V ) )
52, 4syl 17 . . . 4  |-  ( Fun 
F  ->  ( A F y  ->  A  e.  _V ) )
65ancrd 563 . . 3  |-  ( Fun 
F  ->  ( A F y  ->  ( A  e.  _V  /\  A F y ) ) )
76alrimiv 1781 . 2  |-  ( Fun 
F  ->  A. y
( A F y  ->  ( A  e. 
_V  /\  A F
y ) ) )
8 breq1 4398 . . . . . . 7  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
98mobidv 2340 . . . . . 6  |-  ( x  =  A  ->  ( E* y  x F
y  <->  E* y  A F y ) )
109imbi2d 323 . . . . 5  |-  ( x  =  A  ->  (
( Fun  F  ->  E* y  x F y )  <->  ( Fun  F  ->  E* y  A F y ) ) )
111simprbi 471 . . . . . 6  |-  ( Fun 
F  ->  A. x E* y  x F
y )
121119.21bi 1967 . . . . 5  |-  ( Fun 
F  ->  E* y  x F y )
1310, 12vtoclg 3093 . . . 4  |-  ( A  e.  _V  ->  ( Fun  F  ->  E* y  A F y ) )
1413com12 31 . . 3  |-  ( Fun 
F  ->  ( A  e.  _V  ->  E* y  A F y ) )
15 moanimv 2380 . . 3  |-  ( E* y ( A  e. 
_V  /\  A F
y )  <->  ( A  e.  _V  ->  E* y  A F y ) )
1614, 15sylibr 217 . 2  |-  ( Fun 
F  ->  E* y
( A  e.  _V  /\  A F y ) )
17 moim 2368 . 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 61 1  |-  ( Fun 
F  ->  E* y  A F y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376   A.wal 1450    = wceq 1452    e. wcel 1904   E*wmo 2320   _Vcvv 3031   class class class wbr 4395   Rel wrel 4844   Fun wfun 5583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-fun 5591
This theorem is referenced by:  funeu  5613  funco  5627  fununmo  5632  imadif  5668  fneu  5690  dff3  6050  shftfn  13213
  Copyright terms: Public domain W3C validator