MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgrancvvdeqlem2 Structured version   Unicode version

Theorem frgrancvvdeqlem2 25735
Description: Lemma 2 for frgrancvvdeq 25746. (Contributed by Alexander van der Vekens, 23-Dec-2017.)
Hypotheses
Ref Expression
frgrancvvdeq.nx  |-  D  =  ( <. V ,  E >. Neighbors  X )
frgrancvvdeq.ny  |-  N  =  ( <. V ,  E >. Neighbors  Y )
frgrancvvdeq.x  |-  ( ph  ->  X  e.  V )
frgrancvvdeq.y  |-  ( ph  ->  Y  e.  V )
frgrancvvdeq.ne  |-  ( ph  ->  X  =/=  Y )
frgrancvvdeq.xy  |-  ( ph  ->  Y  e/  D )
frgrancvvdeq.f  |-  ( ph  ->  V FriendGrph  E )
frgrancvvdeq.a  |-  A  =  ( x  e.  D  |->  ( iota_ y  e.  N  { x ,  y }  e.  ran  E
) )
Assertion
Ref Expression
frgrancvvdeqlem2  |-  ( ph  ->  X  e/  N )
Distinct variable groups:    y, D    x, y, V    x, E, y    y, Y    ph, y
Allowed substitution hints:    ph( x)    A( x, y)    D( x)    N( x, y)    X( x, y)    Y( x)

Proof of Theorem frgrancvvdeqlem2
StepHypRef Expression
1 frgrancvvdeq.f . . . . . 6  |-  ( ph  ->  V FriendGrph  E )
2 frisusgra 25696 . . . . . 6  |-  ( V FriendGrph  E  ->  V USGrph  E )
3 nbgraeledg 25134 . . . . . 6  |-  ( V USGrph  E  ->  ( X  e.  ( <. V ,  E >. Neighbors  Y )  <->  { X ,  Y }  e.  ran  E ) )
41, 2, 33syl 18 . . . . 5  |-  ( ph  ->  ( X  e.  (
<. V ,  E >. Neighbors  Y
)  <->  { X ,  Y }  e.  ran  E ) )
5 frgrancvvdeq.xy . . . . . 6  |-  ( ph  ->  Y  e/  D )
6 df-nel 2619 . . . . . . . . 9  |-  ( Y  e/  D  <->  -.  Y  e.  D )
7 frgrancvvdeq.nx . . . . . . . . . 10  |-  D  =  ( <. V ,  E >. Neighbors  X )
87eleq2i 2498 . . . . . . . . 9  |-  ( Y  e.  D  <->  Y  e.  ( <. V ,  E >. Neighbors  X ) )
96, 8xchbinx 311 . . . . . . . 8  |-  ( Y  e/  D  <->  -.  Y  e.  ( <. V ,  E >. Neighbors  X ) )
10 nbgraeledg 25134 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( Y  e.  ( <. V ,  E >. Neighbors  X )  <->  { Y ,  X }  e.  ran  E ) )
111, 2, 103syl 18 . . . . . . . . 9  |-  ( ph  ->  ( Y  e.  (
<. V ,  E >. Neighbors  X
)  <->  { Y ,  X }  e.  ran  E ) )
1211notbid 295 . . . . . . . 8  |-  ( ph  ->  ( -.  Y  e.  ( <. V ,  E >. Neighbors  X )  <->  -.  { Y ,  X }  e.  ran  E ) )
139, 12syl5bb 260 . . . . . . 7  |-  ( ph  ->  ( Y  e/  D  <->  -. 
{ Y ,  X }  e.  ran  E ) )
14 prcom 4072 . . . . . . . . 9  |-  { Y ,  X }  =  { X ,  Y }
1514eleq1i 2497 . . . . . . . 8  |-  ( { Y ,  X }  e.  ran  E  <->  { X ,  Y }  e.  ran  E )
16 pm2.21 111 . . . . . . . 8  |-  ( -. 
{ X ,  Y }  e.  ran  E  -> 
( { X ,  Y }  e.  ran  E  ->  X  e/  ( <. V ,  E >. Neighbors  Y
) ) )
1715, 16sylnbi 307 . . . . . . 7  |-  ( -. 
{ Y ,  X }  e.  ran  E  -> 
( { X ,  Y }  e.  ran  E  ->  X  e/  ( <. V ,  E >. Neighbors  Y
) ) )
1813, 17syl6bi 231 . . . . . 6  |-  ( ph  ->  ( Y  e/  D  ->  ( { X ,  Y }  e.  ran  E  ->  X  e/  ( <. V ,  E >. Neighbors  Y
) ) ) )
195, 18mpd 15 . . . . 5  |-  ( ph  ->  ( { X ,  Y }  e.  ran  E  ->  X  e/  ( <. V ,  E >. Neighbors  Y
) ) )
204, 19sylbid 218 . . . 4  |-  ( ph  ->  ( X  e.  (
<. V ,  E >. Neighbors  Y
)  ->  X  e/  ( <. V ,  E >. Neighbors  Y ) ) )
2120com12 32 . . 3  |-  ( X  e.  ( <. V ,  E >. Neighbors  Y )  ->  ( ph  ->  X  e/  ( <. V ,  E >. Neighbors  Y
) ) )
22 df-nel 2619 . . . 4  |-  ( X  e/  ( <. V ,  E >. Neighbors  Y )  <->  -.  X  e.  ( <. V ,  E >. Neighbors  Y ) )
23 ax-1 6 . . . 4  |-  ( X  e/  ( <. V ,  E >. Neighbors  Y )  ->  ( ph  ->  X  e/  ( <. V ,  E >. Neighbors  Y
) ) )
2422, 23sylbir 216 . . 3  |-  ( -.  X  e.  ( <. V ,  E >. Neighbors  Y
)  ->  ( ph  ->  X  e/  ( <. V ,  E >. Neighbors  Y
) ) )
2521, 24pm2.61i 167 . 2  |-  ( ph  ->  X  e/  ( <. V ,  E >. Neighbors  Y
) )
26 eqidd 2421 . . 3  |-  ( ph  ->  X  =  X )
27 frgrancvvdeq.ny . . . 4  |-  N  =  ( <. V ,  E >. Neighbors  Y )
2827a1i 11 . . 3  |-  ( ph  ->  N  =  ( <. V ,  E >. Neighbors  Y
) )
2926, 28neleq12d 2760 . 2  |-  ( ph  ->  ( X  e/  N  <->  X  e/  ( <. V ,  E >. Neighbors  Y ) ) )
3025, 29mpbird 235 1  |-  ( ph  ->  X  e/  N )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1867    =/= wne 2616    e/ wnel 2617   {cpr 3995   <.cop 3999   class class class wbr 4417    |-> cmpt 4476   ran crn 4847   iota_crio 6258  (class class class)co 6297   USGrph cusg 25034   Neighbors cnbgra 25121   FriendGrph cfrgra 25692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589  ax-cnex 9591  ax-resscn 9592  ax-1cn 9593  ax-icn 9594  ax-addcl 9595  ax-addrcl 9596  ax-mulcl 9597  ax-mulrcl 9598  ax-mulcom 9599  ax-addass 9600  ax-mulass 9601  ax-distr 9602  ax-i2m1 9603  ax-1ne0 9604  ax-1rid 9605  ax-rnegex 9606  ax-rrecex 9607  ax-cnre 9608  ax-pre-lttri 9609  ax-pre-lttrn 9610  ax-pre-ltadd 9611  ax-pre-mulgt0 9612
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4477  df-mpt 4478  df-tr 4513  df-eprel 4757  df-id 4761  df-po 4767  df-so 4768  df-fr 4805  df-we 4807  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-pred 5391  df-ord 5437  df-on 5438  df-lim 5439  df-suc 5440  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6259  df-ov 6300  df-oprab 6301  df-mpt2 6302  df-om 6699  df-1st 6799  df-2nd 6800  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8370  df-pnf 9673  df-mnf 9674  df-xr 9675  df-ltxr 9676  df-le 9677  df-sub 9858  df-neg 9859  df-nn 10606  df-2 10664  df-n0 10866  df-z 10934  df-uz 11156  df-fz 11779  df-hash 12509  df-usgra 25037  df-nbgra 25124  df-frgra 25693
This theorem is referenced by:  frgrancvvdeqlemA  25741  frgrancvvdeqlemB  25742  frgrancvvdeqlemC  25743
  Copyright terms: Public domain W3C validator