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Theorem frgrancvvdeqlem6 25156
Description: Lemma 6 for frgrancvvdeq 25163. 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 459 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  x  e.  D )
2 riotaex 6162 . . . 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 5865 . . . 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 660 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( A `  x )  =  ( iota_ y  e.  N  { x ,  y }  e.  ran  E ) )
65sneqd 3956 . 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 25154 . 2  |-  ( (
ph  /\  x  e.  D )  ->  { (
iota_ y  e.  N  { x ,  y }  e.  ran  E
) }  =  ( ( <. V ,  E >. Neighbors  x )  i^i  N
) )
156, 14eqtrd 2423 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 367    = wceq 1399    e. wcel 1826    =/= wne 2577    e/ wnel 2578   _Vcvv 3034    i^i cin 3388   {csn 3944   {cpr 3946   <.cop 3950   class class class wbr 4367    |-> cmpt 4425   ran crn 4914   ` cfv 5496   iota_crio 6157  (class class class)co 6196   Neighbors cnbgra 24538   FriendGrph cfrgra 25109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-hash 12308  df-usgra 24454  df-nbgra 24541  df-frgra 25110
This theorem is referenced by:  frgrancvvdeqlem7  25157  frgrancvvdeqlemA  25158  frgrancvvdeqlemC  25160
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