Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funline Structured version   Visualization version   Unicode version

Theorem funline 30980
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 2944 . . . . . 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 2492 . . . . . . . . 9  |-  ( ( l  =  [ <. a ,  b >. ] `'  Colinear  /\  k  =  [ <. a ,  b >. ] `'  Colinear  )  ->  l  =  k )
32ad2ant2l 760 . . . . . . . 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 2876 . . . . . 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 218 . . . . 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 1678 . . . 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 2475 . . . . . . . 8  |-  ( l  =  k  ->  (
l  =  [ <. a ,  b >. ] `'  Colinear  <->  k  =  [ <. a ,  b
>. ] `'  Colinear  ) )
98anbi2d 718 . . . . . . 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 2892 . . . . . 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 5879 . . . . . . . . . 10  |-  ( n  =  m  ->  ( EE `  n )  =  ( EE `  m
) )
1211eleq2d 2534 . . . . . . . . 9  |-  ( n  =  m  ->  (
a  e.  ( EE
`  n )  <->  a  e.  ( EE `  m ) ) )
1311eleq2d 2534 . . . . . . . . 9  |-  ( n  =  m  ->  (
b  e.  ( EE
`  n )  <->  b  e.  ( EE `  m ) ) )
1412, 133anbi12d 1366 . . . . . . . 8  |-  ( n  =  m  ->  (
( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  <->  ( a  e.  ( EE `  m
)  /\  b  e.  ( EE `  m )  /\  a  =/=  b
) ) )
1514anbi1d 719 . . . . . . 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 3006 . . . . . 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 269 . . . . 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 2366 . . . 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 214 . . 3  |-  E* l E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )
2019funoprab 6415 . 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 30975 . . 3  |- Line  =  { <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) }
2221funeqi 5609 . 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 214 1  |-  Fun Line
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007   A.wal 1450    = wceq 1452    e. wcel 1904   E*wmo 2320    =/= wne 2641   E.wrex 2757   <.cop 3965   `'ccnv 4838   Fun wfun 5583   ` cfv 5589   {coprab 6309   [cec 7379   NNcn 10631   EEcee 24997    Colinear ccolin 30875  Linecline2 30972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-iota 5553  df-fun 5591  df-fv 5597  df-oprab 6312  df-line2 30975
This theorem is referenced by:  fvline  30982
  Copyright terms: Public domain W3C validator