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Theorem funline 24765
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 2707 . . . . . 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 2302 . . . . . . . . 9  |-  ( ( l  =  [ <. a ,  b >. ] `'  Colinear  /\  k  =  [ <. a ,  b >. ] `'  Colinear  )  ->  l  =  k )
32ad2ant2l 726 . . . . . . . 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 10 . . . . . . 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 2672 . . . . . 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 204 . . . . 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 1534 . . . 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 2289 . . . . . . . 8  |-  ( l  =  k  ->  (
l  =  [ <. a ,  b >. ] `'  Colinear  <->  k  =  [ <. a ,  b
>. ] `'  Colinear  ) )
98anbi2d 684 . . . . . . 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 2564 . . . . . 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 5525 . . . . . . . . . 10  |-  ( n  =  m  ->  ( EE `  n )  =  ( EE `  m
) )
1211eleq2d 2350 . . . . . . . . 9  |-  ( n  =  m  ->  (
a  e.  ( EE
`  n )  <->  a  e.  ( EE `  m ) ) )
1311eleq2d 2350 . . . . . . . . 9  |-  ( n  =  m  ->  (
b  e.  ( EE
`  n )  <->  b  e.  ( EE `  m ) ) )
1412, 133anbi12d 1253 . . . . . . . 8  |-  ( n  =  m  ->  (
( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  <->  ( a  e.  ( EE `  m
)  /\  b  e.  ( EE `  m )  /\  a  =/=  b
) ) )
1514anbi1d 685 . . . . . . 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 2765 . . . . . 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 252 . . . . 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 2176 . . . 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 200 . . 3  |-  E* l E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )
2019funoprab 5944 . 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 24760 . . 3  |- Line  =  { <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) }
2221funeqi 5275 . 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 200 1  |-  Fun Line
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   A.wal 1527    = wceq 1623    e. wcel 1684   E*wmo 2144    =/= wne 2446   E.wrex 2544   <.cop 3643   `'ccnv 4688   Fun wfun 5249   ` cfv 5255   {coprab 5859   [cec 6658   NNcn 9746   EEcee 24516    Colinear ccolin 24660  Linecline2 24757
This theorem is referenced by:  fvline  24767
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-rex 2549  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-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-iota 5219  df-fun 5257  df-fv 5263  df-oprab 5862  df-line2 24760
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