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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 PPoints
nds.2
Assertion
Ref Expression
nds Ibg Segments

Proof of Theorem nds
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fvex 5539 . . . 4
2 imaexg 5026 . . . 4 PPoints PPoints
31, 2mp1i 11 . . 3 Ibg PPoints PPoints
4 fveq2 5525 . . . . . 6
54imaeq1d 5011 . . . . 5 PPoints PPoints PPoints PPoints
6 fveq2 5525 . . . . . . . 8 PPoints PPoints
76, 6xpeq12d 4714 . . . . . . 7 PPoints PPoints PPoints PPoints
87difeq1d 3293 . . . . . 6 PPoints PPoints PPoints PPoints
98imaeq2d 5012 . . . . 5 PPoints PPoints PPoints PPoints
105, 9eqtrd 2315 . . . 4 PPoints PPoints PPoints PPoints
11 df-Seg 26149 . . . 4 Segments Ibg PPoints PPoints
1210, 11fvmptg 5600 . . 3 Ibg PPoints PPoints Segments PPoints PPoints
133, 12mpdan 649 . 2 Ibg Segments PPoints PPoints
14 nds.2 . . . . 5
1514eqcomi 2287 . . . 4
1615imaeq1i 5009 . . 3 PPoints PPoints PPoints PPoints
17 nds.1 . . . . . . 7 PPoints
1817eqcomi 2287 . . . . . 6 PPoints
1918, 18xpeq12i 4711 . . . . 5 PPoints PPoints
2019difeq1i 3290 . . . 4 PPoints PPoints
2120imaeq2i 5010 . . 3 PPoints PPoints
2216, 21eqtri 2303 . 2 PPoints PPoints
2313, 22syl6eq 2331 1 Ibg Segments
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1623   wcel 1684  cvv 2788   cdif 3149   cid 4304   cxp 4687  cima 4692  cfv 5255  PPointscpoints 26056  Ibgcibg 26107  cseg 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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