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Theorem funline 28178
Description: Show that the Line relationship is a function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
funline  |-  Fun Line

Proof of Theorem funline
Dummy variables  a 
b  k  l  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reeanv 2893 . . . . . 6  |-  ( E. n  e.  NN  E. m  e.  NN  (
( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )  /\  ( ( a  e.  ( EE `  m
)  /\  b  e.  ( EE `  m )  /\  a  =/=  b
)  /\  k  =  [ <. a ,  b
>. ] `'  Colinear  ) )  <-> 
( E. n  e.  NN  ( ( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n
)  /\  a  =/=  b )  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )  /\  E. m  e.  NN  (
( a  e.  ( EE `  m )  /\  b  e.  ( EE `  m )  /\  a  =/=  b
)  /\  k  =  [ <. a ,  b
>. ] `'  Colinear  ) ) )
2 eqtr3 2462 . . . . . . . . 9  |-  ( ( l  =  [ <. a ,  b >. ] `'  Colinear  /\  k  =  [ <. a ,  b >. ] `'  Colinear  )  ->  l  =  k )
32ad2ant2l 745 . . . . . . . 8  |-  ( ( ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )  /\  ( ( a  e.  ( EE `  m
)  /\  b  e.  ( EE `  m )  /\  a  =/=  b
)  /\  k  =  [ <. a ,  b
>. ] `'  Colinear  ) )  ->  l  =  k )
43a1i 11 . . . . . . 7  |-  ( ( n  e.  NN  /\  m  e.  NN )  ->  ( ( ( ( a  e.  ( EE
`  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b )  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )  /\  ( ( a  e.  ( EE `  m
)  /\  b  e.  ( EE `  m )  /\  a  =/=  b
)  /\  k  =  [ <. a ,  b
>. ] `'  Colinear  ) )  ->  l  =  k ) )
54rexlimivv 2851 . . . . . 6  |-  ( E. n  e.  NN  E. m  e.  NN  (
( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )  /\  ( ( a  e.  ( EE `  m
)  /\  b  e.  ( EE `  m )  /\  a  =/=  b
)  /\  k  =  [ <. a ,  b
>. ] `'  Colinear  ) )  ->  l  =  k )
61, 5sylbir 213 . . . . 5  |-  ( ( E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )  /\  E. m  e.  NN  (
( a  e.  ( EE `  m )  /\  b  e.  ( EE `  m )  /\  a  =/=  b
)  /\  k  =  [ <. a ,  b
>. ] `'  Colinear  ) )  ->  l  =  k )
76gen2 1592 . . . 4  |-  A. l A. k ( ( E. n  e.  NN  (
( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )  /\  E. m  e.  NN  (
( a  e.  ( EE `  m )  /\  b  e.  ( EE `  m )  /\  a  =/=  b
)  /\  k  =  [ <. a ,  b
>. ] `'  Colinear  ) )  ->  l  =  k )
8 eqeq1 2449 . . . . . . . 8  |-  ( l  =  k  ->  (
l  =  [ <. a ,  b >. ] `'  Colinear  <->  k  =  [ <. a ,  b
>. ] `'  Colinear  ) )
98anbi2d 703 . . . . . . 7  |-  ( l  =  k  ->  (
( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )  <->  ( (
a  e.  ( EE
`  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b )  /\  k  =  [ <. a ,  b
>. ] `'  Colinear  ) ) )
109rexbidv 2741 . . . . . 6  |-  ( l  =  k  ->  ( E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )  <->  E. n  e.  NN  ( ( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n
)  /\  a  =/=  b )  /\  k  =  [ <. a ,  b
>. ] `'  Colinear  ) ) )
11 fveq2 5696 . . . . . . . . . 10  |-  ( n  =  m  ->  ( EE `  n )  =  ( EE `  m
) )
1211eleq2d 2510 . . . . . . . . 9  |-  ( n  =  m  ->  (
a  e.  ( EE
`  n )  <->  a  e.  ( EE `  m ) ) )
1311eleq2d 2510 . . . . . . . . 9  |-  ( n  =  m  ->  (
b  e.  ( EE
`  n )  <->  b  e.  ( EE `  m ) ) )
1412, 133anbi12d 1290 . . . . . . . 8  |-  ( n  =  m  ->  (
( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  <->  ( a  e.  ( EE `  m
)  /\  b  e.  ( EE `  m )  /\  a  =/=  b
) ) )
1514anbi1d 704 . . . . . . 7  |-  ( n  =  m  ->  (
( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  k  =  [ <. a ,  b
>. ] `'  Colinear  )  <->  ( (
a  e.  ( EE
`  m )  /\  b  e.  ( EE `  m )  /\  a  =/=  b )  /\  k  =  [ <. a ,  b
>. ] `'  Colinear  ) ) )
1615cbvrexv 2953 . . . . . 6  |-  ( E. n  e.  NN  (
( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  k  =  [ <. a ,  b
>. ] `'  Colinear  )  <->  E. m  e.  NN  ( ( a  e.  ( EE `  m )  /\  b  e.  ( EE `  m
)  /\  a  =/=  b )  /\  k  =  [ <. a ,  b
>. ] `'  Colinear  ) )
1710, 16syl6bb 261 . . . . 5  |-  ( l  =  k  ->  ( E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )  <->  E. m  e.  NN  ( ( a  e.  ( EE `  m )  /\  b  e.  ( EE `  m
)  /\  a  =/=  b )  /\  k  =  [ <. a ,  b
>. ] `'  Colinear  ) ) )
1817mo4 2317 . . . 4  |-  ( E* l E. n  e.  NN  ( ( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n
)  /\  a  =/=  b )  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )  <->  A. l A. k ( ( E. n  e.  NN  (
( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )  /\  E. m  e.  NN  (
( a  e.  ( EE `  m )  /\  b  e.  ( EE `  m )  /\  a  =/=  b
)  /\  k  =  [ <. a ,  b
>. ] `'  Colinear  ) )  ->  l  =  k ) )
197, 18mpbir 209 . . 3  |-  E* l E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )
2019funoprab 6195 . 2  |-  Fun  { <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) }
21 df-line2 28173 . . 3  |- Line  =  { <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) }
2221funeqi 5443 . 2  |-  ( Fun Line  <->  Fun 
{ <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) } )
2320, 22mpbir 209 1  |-  Fun Line
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369    e. wcel 1756   E*wmo 2254    =/= wne 2611   E.wrex 2721   <.cop 3888   `'ccnv 4844   Fun wfun 5417   ` cfv 5423   {coprab 6097   [cec 7104   NNcn 10327   EEcee 23139    Colinear ccolin 28073  Linecline2 28170
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 4418  ax-nul 4426  ax-pr 4536
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-iota 5386  df-fun 5425  df-fv 5431  df-oprab 6100  df-line2 28173
This theorem is referenced by:  fvline  28180
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