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Theorem iscola2 26092
Description: The predicate "being collinear points". (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Hypothesis
Ref Expression
iscola2.1  |-  ( ph  ->  G  e. Ig )
Assertion
Ref Expression
iscola2  |-  ( ph  ->  (coln `  G )  =  { x  |  E. l  e.  (PLines `  G
) x  C_  l } )
Distinct variable group:    x, l, G
Allowed substitution hints:    ph( x, l)

Proof of Theorem iscola2
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 iscola2.1 . 2  |-  ( ph  ->  G  e. Ig )
2 iunab 3948 . . 3  |-  U_ l  e.  (PLines `  G ) { x  |  x  C_  l }  =  {
x  |  E. l  e.  (PLines `  G )
x  C_  l }
3 fvex 5539 . . . . 5  |-  (PLines `  G )  e.  _V
43a1i 10 . . . 4  |-  ( ph  ->  (PLines `  G )  e.  _V )
5 pm4.24 624 . . . . . . 7  |-  ( x 
C_  l  <->  ( x  C_  l  /\  x  C_  l ) )
65abbii 2395 . . . . . 6  |-  { x  |  x  C_  l }  =  { x  |  ( x  C_  l  /\  x  C_  l ) }
7 abssexg 4195 . . . . . 6  |-  ( l  e.  (PLines `  G
)  ->  { x  |  ( x  C_  l  /\  x  C_  l
) }  e.  _V )
86, 7syl5eqel 2367 . . . . 5  |-  ( l  e.  (PLines `  G
)  ->  { x  |  x  C_  l }  e.  _V )
98rgen 2608 . . . 4  |-  A. l  e.  (PLines `  G ) { x  |  x  C_  l }  e.  _V
10 iunexg 5767 . . . 4  |-  ( ( (PLines `  G )  e.  _V  /\  A. l  e.  (PLines `  G ) { x  |  x  C_  l }  e.  _V )  ->  U_ l  e.  (PLines `  G ) { x  |  x  C_  l }  e.  _V )
114, 9, 10sylancl 643 . . 3  |-  ( ph  ->  U_ l  e.  (PLines `  G ) { x  |  x  C_  l }  e.  _V )
122, 11syl5eqelr 2368 . 2  |-  ( ph  ->  { x  |  E. l  e.  (PLines `  G
) x  C_  l }  e.  _V )
13 fveq2 5525 . . . . 5  |-  ( f  =  G  ->  (PLines `  f )  =  (PLines `  G ) )
1413rexeqdv 2743 . . . 4  |-  ( f  =  G  ->  ( E. l  e.  (PLines `  f ) x  C_  l 
<->  E. l  e.  (PLines `  G ) x  C_  l ) )
1514abbidv 2397 . . 3  |-  ( f  =  G  ->  { x  |  E. l  e.  (PLines `  f ) x  C_  l }  =  {
x  |  E. l  e.  (PLines `  G )
x  C_  l }
)
16 df-col 26091 . . 3  |- coln  =  ( f  e. Ig  |->  { x  |  E. l  e.  (PLines `  f ) x  C_  l } )
1715, 16fvmptg 5600 . 2  |-  ( ( G  e. Ig  /\  {
x  |  E. l  e.  (PLines `  G )
x  C_  l }  e.  _V )  ->  (coln `  G )  =  {
x  |  E. l  e.  (PLines `  G )
x  C_  l }
)
181, 12, 17syl2anc 642 1  |-  ( ph  ->  (coln `  G )  =  { x  |  E. l  e.  (PLines `  G
) x  C_  l } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   U_ciun 3905   ` cfv 5255  PLinescplines 26058  Igcig 26060  colnccol 26090
This theorem is referenced by:  iscol2  26093
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-col 26091
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