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Theorem fununiq 25151
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 5405 . 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 4159 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x F y  <-> 
A F B ) )
653adant3 977 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x F y  <-> 
A F B ) )
7 breq12 4159 . . . . . . . 8  |-  ( ( x  =  A  /\  z  =  C )  ->  ( x F z  <-> 
A F C ) )
873adant2 976 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x F z  <-> 
A F C ) )
96, 8anbi12d 692 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( x F y  /\  x F z )  <->  ( A F B  /\  A F C ) ) )
10 eqeq12 2400 . . . . . . 7  |-  ( ( y  =  B  /\  z  =  C )  ->  ( y  =  z  <-> 
B  =  C ) )
11103adant1 975 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( y  =  z  <-> 
B  =  C ) )
129, 11imbi12d 312 . . . . 5  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( ( x F y  /\  x F z )  -> 
y  =  z )  <-> 
( ( A F B  /\  A F C )  ->  B  =  C ) ) )
1312spc3gv 2985 . . . 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 1279 . . 3  |-  ( A. x A. y A. z
( ( x F y  /\  x F z )  ->  y  =  z )  -> 
( ( A F B  /\  A F C )  ->  B  =  C ) )
1514adantl 453 . 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 188 1  |-  ( Fun 
F  ->  ( ( A F B  /\  A F C )  ->  B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1546    = wceq 1649    e. wcel 1717   _Vcvv 2900   class class class wbr 4154   Rel wrel 4824   Fun wfun 5389
This theorem is referenced by:  funbreq  25152
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 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-id 4440  df-cnv 4827  df-co 4828  df-fun 5397
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