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Theorem frgrancvvdeqlem6 24859
Description: Lemma 6 for frgrancvvdeq 24866. 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 6260 . . . 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 5964 . . . 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 4045 . 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 24857 . 2  |-  ( (
ph  /\  x  e.  D )  ->  { (
iota_ y  e.  N  { x ,  y }  e.  ran  E
) }  =  ( ( <. V ,  E >. Neighbors  x )  i^i  N
) )
156, 14eqtrd 2508 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 1379    e. wcel 1767    =/= wne 2662    e/ wnel 2663   _Vcvv 3118    i^i cin 3480   {csn 4033   {cpr 4035   <.cop 4039   class class class wbr 4453    |-> cmpt 4511   ran crn 5006   ` cfv 5594   iota_crio 6255  (class class class)co 6295   Neighbors cnbgra 24240   FriendGrph cfrgra 24811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-hash 12386  df-usgra 24156  df-nbgra 24243  df-frgra 24812
This theorem is referenced by:  frgrancvvdeqlem7  24860  frgrancvvdeqlemA  24861  frgrancvvdeqlemC  24863
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