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Theorem nds 26253
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 5555 . . . 4  |-  ( seg `  G )  e.  _V
2 imaexg 5042 . . . 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 5541 . . . . . 6  |-  ( f  =  G  ->  ( seg `  f )  =  ( seg `  G
) )
54imaeq1d 5027 . . . . 5  |-  ( f  =  G  ->  (
( seg `  f
) " ( ( (PPoints `  f )  X.  (PPoints `  f )
)  \  _I  )
)  =  ( ( seg `  G )
" ( ( (PPoints `  f )  X.  (PPoints `  f ) )  \  _I  ) ) )
6 fveq2 5541 . . . . . . . 8  |-  ( f  =  G  ->  (PPoints `  f )  =  (PPoints `  G ) )
76, 6xpeq12d 4730 . . . . . . 7  |-  ( f  =  G  ->  (
(PPoints `  f )  X.  (PPoints `  f )
)  =  ( (PPoints `  G )  X.  (PPoints `  G ) ) )
87difeq1d 3306 . . . . . 6  |-  ( f  =  G  ->  (
( (PPoints `  f
)  X.  (PPoints `  f
) )  \  _I  )  =  ( (
(PPoints `  G )  X.  (PPoints `  G )
)  \  _I  )
)
98imaeq2d 5028 . . . . 5  |-  ( f  =  G  ->  (
( seg `  G
) " ( ( (PPoints `  f )  X.  (PPoints `  f )
)  \  _I  )
)  =  ( ( seg `  G )
" ( ( (PPoints `  G )  X.  (PPoints `  G ) )  \  _I  ) ) )
105, 9eqtrd 2328 . . . 4  |-  ( f  =  G  ->  (
( seg `  f
) " ( ( (PPoints `  f )  X.  (PPoints `  f )
)  \  _I  )
)  =  ( ( seg `  G )
" ( ( (PPoints `  G )  X.  (PPoints `  G ) )  \  _I  ) ) )
11 df-Seg 26252 . . . 4  |- Segments  =  ( f  e. Ibg  |->  ( ( seg `  f )
" ( ( (PPoints `  f )  X.  (PPoints `  f ) )  \  _I  ) ) )
1210, 11fvmptg 5616 . . 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 2300 . . . 4  |-  ( seg `  G )  =  S
1615imaeq1i 5025 . . 3  |-  ( ( seg `  G )
" ( ( (PPoints `  G )  X.  (PPoints `  G ) )  \  _I  ) )  =  ( S " ( ( (PPoints `  G )  X.  (PPoints `  G )
)  \  _I  )
)
17 nds.1 . . . . . . 7  |-  P  =  (PPoints `  G )
1817eqcomi 2300 . . . . . 6  |-  (PPoints `  G
)  =  P
1918, 18xpeq12i 4727 . . . . 5  |-  ( (PPoints `  G )  X.  (PPoints `  G ) )  =  ( P  X.  P
)
2019difeq1i 3303 . . . 4  |-  ( ( (PPoints `  G )  X.  (PPoints `  G )
)  \  _I  )  =  ( ( P  X.  P )  \  _I  )
2120imaeq2i 5026 . . 3  |-  ( S
" ( ( (PPoints `  G )  X.  (PPoints `  G ) )  \  _I  ) )  =  ( S " ( ( P  X.  P ) 
\  _I  ) )
2216, 21eqtri 2316 . 2  |-  ( ( seg `  G )
" ( ( (PPoints `  G )  X.  (PPoints `  G ) )  \  _I  ) )  =  ( S " ( ( P  X.  P ) 
\  _I  ) )
2313, 22syl6eq 2344 1  |-  ( G  e. Ibg  ->  (Segments `  G )  =  ( S "
( ( P  X.  P )  \  _I  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    _I cid 4320    X. cxp 4703   "cima 4708   ` cfv 5271  PPointscpoints 26159  Ibgcibg 26210   segcseg 26233  SegmentscSeg 26251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-Seg 26252
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