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Theorem nds 26150
Description: The non-degenerated segments. (For my private use only. Don't use.) (Contributed by FL, 7-Mar-2016.)
Hypotheses
Ref Expression
nds.1  |-  P  =  (PPoints `  G )
nds.2  |-  S  =  ( seg `  G
)
Assertion
Ref Expression
nds  |-  ( G  e. Ibg  ->  (Segments `  G )  =  ( S "
( ( P  X.  P )  \  _I  ) ) )

Proof of Theorem nds
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 fvex 5539 . . . 4  |-  ( seg `  G )  e.  _V
2 imaexg 5026 . . . 4  |-  ( ( seg `  G )  e.  _V  ->  (
( seg `  G
) " ( ( (PPoints `  G )  X.  (PPoints `  G )
)  \  _I  )
)  e.  _V )
31, 2mp1i 11 . . 3  |-  ( G  e. Ibg  ->  ( ( seg `  G ) " (
( (PPoints `  G
)  X.  (PPoints `  G
) )  \  _I  ) )  e.  _V )
4 fveq2 5525 . . . . . 6  |-  ( f  =  G  ->  ( seg `  f )  =  ( seg `  G
) )
54imaeq1d 5011 . . . . 5  |-  ( f  =  G  ->  (
( seg `  f
) " ( ( (PPoints `  f )  X.  (PPoints `  f )
)  \  _I  )
)  =  ( ( seg `  G )
" ( ( (PPoints `  f )  X.  (PPoints `  f ) )  \  _I  ) ) )
6 fveq2 5525 . . . . . . . 8  |-  ( f  =  G  ->  (PPoints `  f )  =  (PPoints `  G ) )
76, 6xpeq12d 4714 . . . . . . 7  |-  ( f  =  G  ->  (
(PPoints `  f )  X.  (PPoints `  f )
)  =  ( (PPoints `  G )  X.  (PPoints `  G ) ) )
87difeq1d 3293 . . . . . 6  |-  ( f  =  G  ->  (
( (PPoints `  f
)  X.  (PPoints `  f
) )  \  _I  )  =  ( (
(PPoints `  G )  X.  (PPoints `  G )
)  \  _I  )
)
98imaeq2d 5012 . . . . 5  |-  ( f  =  G  ->  (
( seg `  G
) " ( ( (PPoints `  f )  X.  (PPoints `  f )
)  \  _I  )
)  =  ( ( seg `  G )
" ( ( (PPoints `  G )  X.  (PPoints `  G ) )  \  _I  ) ) )
105, 9eqtrd 2315 . . . 4  |-  ( f  =  G  ->  (
( seg `  f
) " ( ( (PPoints `  f )  X.  (PPoints `  f )
)  \  _I  )
)  =  ( ( seg `  G )
" ( ( (PPoints `  G )  X.  (PPoints `  G ) )  \  _I  ) ) )
11 df-Seg 26149 . . . 4  |- Segments  =  ( f  e. Ibg  |->  ( ( seg `  f )
" ( ( (PPoints `  f )  X.  (PPoints `  f ) )  \  _I  ) ) )
1210, 11fvmptg 5600 . . 3  |-  ( ( G  e. Ibg  /\  (
( seg `  G
) " ( ( (PPoints `  G )  X.  (PPoints `  G )
)  \  _I  )
)  e.  _V )  ->  (Segments `  G )  =  ( ( seg `  G ) " (
( (PPoints `  G
)  X.  (PPoints `  G
) )  \  _I  ) ) )
133, 12mpdan 649 . 2  |-  ( G  e. Ibg  ->  (Segments `  G )  =  ( ( seg `  G ) " (
( (PPoints `  G
)  X.  (PPoints `  G
) )  \  _I  ) ) )
14 nds.2 . . . . 5  |-  S  =  ( seg `  G
)
1514eqcomi 2287 . . . 4  |-  ( seg `  G )  =  S
1615imaeq1i 5009 . . 3  |-  ( ( seg `  G )
" ( ( (PPoints `  G )  X.  (PPoints `  G ) )  \  _I  ) )  =  ( S " ( ( (PPoints `  G )  X.  (PPoints `  G )
)  \  _I  )
)
17 nds.1 . . . . . . 7  |-  P  =  (PPoints `  G )
1817eqcomi 2287 . . . . . 6  |-  (PPoints `  G
)  =  P
1918, 18xpeq12i 4711 . . . . 5  |-  ( (PPoints `  G )  X.  (PPoints `  G ) )  =  ( P  X.  P
)
2019difeq1i 3290 . . . 4  |-  ( ( (PPoints `  G )  X.  (PPoints `  G )
)  \  _I  )  =  ( ( P  X.  P )  \  _I  )
2120imaeq2i 5010 . . 3  |-  ( S
" ( ( (PPoints `  G )  X.  (PPoints `  G ) )  \  _I  ) )  =  ( S " ( ( P  X.  P ) 
\  _I  ) )
2216, 21eqtri 2303 . 2  |-  ( ( seg `  G )
" ( ( (PPoints `  G )  X.  (PPoints `  G ) )  \  _I  ) )  =  ( S " ( ( P  X.  P ) 
\  _I  ) )
2313, 22syl6eq 2331 1  |-  ( G  e. Ibg  ->  (Segments `  G )  =  ( S "
( ( P  X.  P )  \  _I  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    _I cid 4304    X. cxp 4687   "cima 4692   ` cfv 5255  PPointscpoints 26056  Ibgcibg 26107   segcseg 26130  SegmentscSeg 26148
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-sep 4141  ax-nul 4149  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-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-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-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-Seg 26149
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