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Theorem fununiq 24126
Description: The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)
Hypotheses
Ref Expression
fununiq.1  |-  A  e. 
_V
fununiq.2  |-  B  e. 
_V
fununiq.3  |-  C  e. 
_V
Assertion
Ref Expression
fununiq  |-  ( Fun 
F  ->  ( ( A F B  /\  A F C )  ->  B  =  C ) )

Proof of Theorem fununiq
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffun2 5265 . 2  |-  ( Fun 
F  <->  ( Rel  F  /\  A. x A. y A. z ( ( x F y  /\  x F z )  -> 
y  =  z ) ) )
2 fununiq.1 . . . 4  |-  A  e. 
_V
3 fununiq.2 . . . 4  |-  B  e. 
_V
4 fununiq.3 . . . 4  |-  C  e. 
_V
5 breq12 4028 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x F y  <-> 
A F B ) )
653adant3 975 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x F y  <-> 
A F B ) )
7 breq12 4028 . . . . . . . 8  |-  ( ( x  =  A  /\  z  =  C )  ->  ( x F z  <-> 
A F C ) )
873adant2 974 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x F z  <-> 
A F C ) )
96, 8anbi12d 691 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( x F y  /\  x F z )  <->  ( A F B  /\  A F C ) ) )
10 eqeq12 2295 . . . . . . 7  |-  ( ( y  =  B  /\  z  =  C )  ->  ( y  =  z  <-> 
B  =  C ) )
11103adant1 973 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( y  =  z  <-> 
B  =  C ) )
129, 11imbi12d 311 . . . . 5  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( ( x F y  /\  x F z )  -> 
y  =  z )  <-> 
( ( A F B  /\  A F C )  ->  B  =  C ) ) )
1312spc3gv 2873 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( A. x A. y A. z ( ( x F y  /\  x F z )  -> 
y  =  z )  ->  ( ( A F B  /\  A F C )  ->  B  =  C ) ) )
142, 3, 4, 13mp3an 1277 . . 3  |-  ( A. x A. y A. z
( ( x F y  /\  x F z )  ->  y  =  z )  -> 
( ( A F B  /\  A F C )  ->  B  =  C ) )
1514adantl 452 . 2  |-  ( ( Rel  F  /\  A. x A. y A. z
( ( x F y  /\  x F z )  ->  y  =  z ) )  ->  ( ( A F B  /\  A F C )  ->  B  =  C ) )
161, 15sylbi 187 1  |-  ( Fun 
F  ->  ( ( A F B  /\  A F C )  ->  B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023   Rel wrel 4694   Fun wfun 5249
This theorem is referenced by:  funbreq  24127
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-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-cnv 4697  df-co 4698  df-fun 5257
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