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Theorem fununiq 27580
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 5427 . 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 4296 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x F y  <-> 
A F B ) )
653adant3 1008 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x F y  <-> 
A F B ) )
7 breq12 4296 . . . . . . . 8  |-  ( ( x  =  A  /\  z  =  C )  ->  ( x F z  <-> 
A F C ) )
873adant2 1007 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x F z  <-> 
A F C ) )
96, 8anbi12d 710 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( x F y  /\  x F z )  <->  ( A F B  /\  A F C ) ) )
10 eqeq12 2454 . . . . . . 7  |-  ( ( y  =  B  /\  z  =  C )  ->  ( y  =  z  <-> 
B  =  C ) )
11103adant1 1006 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( y  =  z  <-> 
B  =  C ) )
129, 11imbi12d 320 . . . . 5  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( ( x F y  /\  x F z )  -> 
y  =  z )  <-> 
( ( A F B  /\  A F C )  ->  B  =  C ) ) )
1312spc3gv 3061 . . . 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 1314 . . 3  |-  ( A. x A. y A. z
( ( x F y  /\  x F z )  ->  y  =  z )  -> 
( ( A F B  /\  A F C )  ->  B  =  C ) )
1514adantl 466 . 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 195 1  |-  ( Fun 
F  ->  ( ( A F B  /\  A F C )  ->  B  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369    e. wcel 1756   _Vcvv 2971   class class class wbr 4291   Rel wrel 4844   Fun wfun 5411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-br 4292  df-opab 4350  df-id 4635  df-cnv 4847  df-co 4848  df-fun 5419
This theorem is referenced by:  funbreq  27581
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