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Theorem funline 26068
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 2867 . . . . . 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 2454 . . . . . . . . 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 2827 . . . . . 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 1556 . . . 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 2441 . . . . . . . 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 2718 . . . . . 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 5720 . . . . . . . . . 10  |-  ( n  =  m  ->  ( EE `  n )  =  ( EE `  m
) )
1211eleq2d 2502 . . . . . . . . 9  |-  ( n  =  m  ->  (
a  e.  ( EE
`  n )  <->  a  e.  ( EE `  m ) ) )
1311eleq2d 2502 . . . . . . . . 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 2925 . . . . . 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 2313 . . . 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 6162 . 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 26063 . . 3  |- Line  =  { <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) }
2221funeqi 5466 . 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 1549    = wceq 1652    e. wcel 1725   E*wmo 2281    =/= wne 2598   E.wrex 2698   <.cop 3809   `'ccnv 4869   Fun wfun 5440   ` cfv 5446   {coprab 6074   [cec 6895   NNcn 9992   EEcee 25819    Colinear ccolin 25963  Linecline2 26060
This theorem is referenced by:  fvline  26070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-iota 5410  df-fun 5448  df-fv 5454  df-oprab 6077  df-line2 26063
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