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Theorem frgrancvvdeqlem6 30799
Description: Lemma 6 for frgrancvvdeq 30806. The mapping of neighbors to neighbors applied on a vertex is the intersection of the corresponding neighborhoods. (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
frgrancvvdeqlem6  |-  ( (
ph  /\  x  e.  D )  ->  { ( A `  x ) }  =  ( (
<. V ,  E >. Neighbors  x
)  i^i  N )
)
Distinct variable groups:    y, D, x    x, V, y    x, E, y    y, Y    ph, y    y, N    x, D    x, N    ph, x
Allowed substitution hints:    A( x, y)    X( x, y)    Y( x)

Proof of Theorem frgrancvvdeqlem6
StepHypRef Expression
1 simpr 461 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  x  e.  D )
2 riotaex 6168 . . . 4  |-  ( iota_ y  e.  N  { x ,  y }  e.  ran  E )  e.  _V
3 frgrancvvdeq.a . . . . 5  |-  A  =  ( x  e.  D  |->  ( iota_ y  e.  N  { x ,  y }  e.  ran  E
) )
43fvmpt2 5893 . . . 4  |-  ( ( x  e.  D  /\  ( iota_ y  e.  N  { x ,  y }  e.  ran  E
)  e.  _V )  ->  ( A `  x
)  =  ( iota_ y  e.  N  { x ,  y }  e.  ran  E ) )
51, 2, 4sylancl 662 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( A `  x )  =  ( iota_ y  e.  N  { x ,  y }  e.  ran  E ) )
65sneqd 4000 . 2  |-  ( (
ph  /\  x  e.  D )  ->  { ( A `  x ) }  =  { (
iota_ y  e.  N  { x ,  y }  e.  ran  E
) } )
7 frgrancvvdeq.nx . . 3  |-  D  =  ( <. V ,  E >. Neighbors  X )
8 frgrancvvdeq.ny . . 3  |-  N  =  ( <. V ,  E >. Neighbors  Y )
9 frgrancvvdeq.x . . 3  |-  ( ph  ->  X  e.  V )
10 frgrancvvdeq.y . . 3  |-  ( ph  ->  Y  e.  V )
11 frgrancvvdeq.ne . . 3  |-  ( ph  ->  X  =/=  Y )
12 frgrancvvdeq.xy . . 3  |-  ( ph  ->  Y  e/  D )
13 frgrancvvdeq.f . . 3  |-  ( ph  ->  V FriendGrph  E )
147, 8, 9, 10, 11, 12, 13, 3frgrancvvdeqlem4 30797 . 2  |-  ( (
ph  /\  x  e.  D )  ->  { (
iota_ y  e.  N  { x ,  y }  e.  ran  E
) }  =  ( ( <. V ,  E >. Neighbors  x )  i^i  N
) )
156, 14eqtrd 2495 1  |-  ( (
ph  /\  x  e.  D )  ->  { ( A `  x ) }  =  ( (
<. V ,  E >. Neighbors  x
)  i^i  N )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648    e/ wnel 2649   _Vcvv 3078    i^i cin 3438   {csn 3988   {cpr 3990   <.cop 3994   class class class wbr 4403    |-> cmpt 4461   ran crn 4952   ` cfv 5529   iota_crio 6163  (class class class)co 6203   Neighbors cnbgra 23508   FriendGrph cfrgra 30751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-hash 12225  df-usgra 23445  df-nbgra 23511  df-frgra 30752
This theorem is referenced by:  frgrancvvdeqlem7  30800  frgrancvvdeqlemA  30801  frgrancvvdeqlemC  30803
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