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Theorem funline 25980
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 2835 . . . . . 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 2423 . . . . . . . . 9  |-  ( ( l  =  [ <. a ,  b >. ] `'  Colinear  /\  k  =  [ <. a ,  b >. ] `'  Colinear  )  ->  l  =  k )
32ad2ant2l 727 . . . . . . . 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 2795 . . . . . 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 205 . . . . 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 1553 . . . 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 2410 . . . . . . . 8  |-  ( l  =  k  ->  (
l  =  [ <. a ,  b >. ] `'  Colinear  <->  k  =  [ <. a ,  b
>. ] `'  Colinear  ) )
98anbi2d 685 . . . . . . 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 2687 . . . . . 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 5687 . . . . . . . . . 10  |-  ( n  =  m  ->  ( EE `  n )  =  ( EE `  m
) )
1211eleq2d 2471 . . . . . . . . 9  |-  ( n  =  m  ->  (
a  e.  ( EE
`  n )  <->  a  e.  ( EE `  m ) ) )
1311eleq2d 2471 . . . . . . . . 9  |-  ( n  =  m  ->  (
b  e.  ( EE
`  n )  <->  b  e.  ( EE `  m ) ) )
1412, 133anbi12d 1255 . . . . . . . 8  |-  ( n  =  m  ->  (
( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  <->  ( a  e.  ( EE `  m
)  /\  b  e.  ( EE `  m )  /\  a  =/=  b
) ) )
1514anbi1d 686 . . . . . . 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 2893 . . . . . 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 253 . . . . 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 2287 . . . 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 201 . . 3  |-  E* l E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )
2019funoprab 6129 . 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 25975 . . 3  |- Line  =  { <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) }
2221funeqi 5433 . 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 201 1  |-  Fun Line
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   A.wal 1546    = wceq 1649    e. wcel 1721   E*wmo 2255    =/= wne 2567   E.wrex 2667   <.cop 3777   `'ccnv 4836   Fun wfun 5407   ` cfv 5413   {coprab 6041   [cec 6862   NNcn 9956   EEcee 25731    Colinear ccolin 25875  Linecline2 25972
This theorem is referenced by:  fvline  25982
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-iota 5377  df-fun 5415  df-fv 5421  df-oprab 6044  df-line2 25975
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