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Theorem isconcl1b 26097
Description: The predicate "are concurrent lines". (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Hypothesis
Ref Expression
isconclb  |-  ( ph  ->  F  e. Ig )
Assertion
Ref Expression
isconcl1b  |-  ( ph  ->  (con `  F )  =  { x  e.  ~P (PLines `  F )  | 
|^| x  =/=  (/) } )
Distinct variable group:    x, F
Allowed substitution hint:    ph( x)

Proof of Theorem isconcl1b
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 isconclb . 2  |-  ( ph  ->  F  e. Ig )
2 fveq2 5525 . . . . 5  |-  ( f  =  F  ->  (PLines `  f )  =  (PLines `  F ) )
32pweqd 3630 . . . 4  |-  ( f  =  F  ->  ~P (PLines `  f )  =  ~P (PLines `  F
) )
4 biidd 228 . . . 4  |-  ( f  =  F  ->  ( |^| x  =/=  (/)  <->  |^| x  =/=  (/) ) )
53, 4rabeqbidv 2783 . . 3  |-  ( f  =  F  ->  { x  e.  ~P (PLines `  f
)  |  |^| x  =/=  (/) }  =  {
x  e.  ~P (PLines `  F )  |  |^| x  =/=  (/) } )
6 df-con2 26096 . . 3  |- con  =  ( f  e. Ig  |->  { x  e.  ~P (PLines `  f
)  |  |^| x  =/=  (/) } )
7 fvex 5539 . . . . 5  |-  (PLines `  F )  e.  _V
87pwex 4193 . . . 4  |-  ~P (PLines `  F )  e.  _V
98rabex 4165 . . 3  |-  { x  e.  ~P (PLines `  F
)  |  |^| x  =/=  (/) }  e.  _V
105, 6, 9fvmpt 5602 . 2  |-  ( F  e. Ig  ->  (con `  F
)  =  { x  e.  ~P (PLines `  F
)  |  |^| x  =/=  (/) } )
111, 10syl 15 1  |-  ( ph  ->  (con `  F )  =  { x  e.  ~P (PLines `  F )  | 
|^| x  =/=  (/) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   (/)c0 3455   ~Pcpw 3625   |^|cint 3862   ` cfv 5255  PLinescplines 26058  Igcig 26060  conccon2 26095
This theorem is referenced by:  isconcl2b  26098
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-pow 4188  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-sbc 2992  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-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-iota 5219  df-fun 5257  df-fv 5263  df-con2 26096
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