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Theorem frgrancvvdeqlem1 25814
Description: Lemma 1 for frgrancvvdeq 25826. (Contributed by Alexander van der Vekens, 22-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
frgrancvvdeqlem1  |-  ( (
ph  /\  x  e.  D )  ->  Y  e.  ( V  \  {
x } ) )
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 frgrancvvdeqlem1
StepHypRef Expression
1 frgrancvvdeq.y . . 3  |-  ( ph  ->  Y  e.  V )
21adantr 471 . 2  |-  ( (
ph  /\  x  e.  D )  ->  Y  e.  V )
3 frgrancvvdeq.xy . . . . 5  |-  ( ph  ->  Y  e/  D )
4 df-nel 2636 . . . . . 6  |-  ( Y  e/  D  <->  -.  Y  e.  D )
5 eleq1a 2535 . . . . . . 7  |-  ( x  e.  D  ->  ( Y  =  x  ->  Y  e.  D ) )
65con3rr3 143 . . . . . 6  |-  ( -.  Y  e.  D  -> 
( x  e.  D  ->  -.  Y  =  x ) )
74, 6sylbi 200 . . . . 5  |-  ( Y  e/  D  ->  (
x  e.  D  ->  -.  Y  =  x
) )
83, 7syl 17 . . . 4  |-  ( ph  ->  ( x  e.  D  ->  -.  Y  =  x ) )
98imp 435 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  -.  Y  =  x )
10 elsncg 4003 . . . . 5  |-  ( Y  e.  V  ->  ( Y  e.  { x } 
<->  Y  =  x ) )
111, 10syl 17 . . . 4  |-  ( ph  ->  ( Y  e.  {
x }  <->  Y  =  x ) )
1211adantr 471 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( Y  e.  { x } 
<->  Y  =  x ) )
139, 12mtbird 307 . 2  |-  ( (
ph  /\  x  e.  D )  ->  -.  Y  e.  { x } )
142, 13eldifd 3427 1  |-  ( (
ph  /\  x  e.  D )  ->  Y  e.  ( V  \  {
x } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633    e/ wnel 2634    \ cdif 3413   {csn 3980   {cpr 3982   <.cop 3986   class class class wbr 4418    |-> cmpt 4477   ran crn 4857   iota_crio 6281  (class class class)co 6320   Neighbors cnbgra 25201   FriendGrph cfrgra 25772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-nel 2636  df-v 3059  df-dif 3419  df-sn 3981
This theorem is referenced by:  frgrancvvdeqlem3  25816  frgrancvvdeqlem4  25817
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