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Theorem fununiq 30361
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 5554 . 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 4371 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x F y  <-> 
A F B ) )
653adant3 1025 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x F y  <-> 
A F B ) )
7 breq12 4371 . . . . . . . 8  |-  ( ( x  =  A  /\  z  =  C )  ->  ( x F z  <-> 
A F C ) )
873adant2 1024 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x F z  <-> 
A F C ) )
96, 8anbi12d 715 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( x F y  /\  x F z )  <->  ( A F B  /\  A F C ) ) )
10 eqeq12 2441 . . . . . . 7  |-  ( ( y  =  B  /\  z  =  C )  ->  ( y  =  z  <-> 
B  =  C ) )
11103adant1 1023 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( y  =  z  <-> 
B  =  C ) )
129, 11imbi12d 321 . . . . 5  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( ( x F y  /\  x F z )  -> 
y  =  z )  <-> 
( ( A F B  /\  A F C )  ->  B  =  C ) ) )
1312spc3gv 3114 . . . 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 1360 . . 3  |-  ( A. x A. y A. z
( ( x F y  /\  x F z )  ->  y  =  z )  -> 
( ( A F B  /\  A F C )  ->  B  =  C ) )
1514adantl 467 . 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 198 1  |-  ( Fun 
F  ->  ( ( A F B  /\  A F C )  ->  B  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437    e. wcel 1872   _Vcvv 3022   class class class wbr 4366   Rel wrel 4801   Fun wfun 5538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-opab 4426  df-id 4711  df-cnv 4804  df-co 4805  df-fun 5546
This theorem is referenced by:  funbreq  30362
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