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Theorem nbgraopALT 25152
Description: Alternate proof of nbgraop 25151 using mpt2xopoveq 6965, but being longer. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
nbgraopALT  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
Distinct variable groups:    n, V    n, E    n, N    n, Y    n, Z

Proof of Theorem nbgraopALT
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nbgra 25148 . . 3  |- Neighbors  =  ( x  e.  _V , 
y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x
)  |  { y ,  n }  e.  ran  ( 2nd `  x
) } )
21mpt2xopoveq 6965 . 2  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  =  { n  e.  V  |  [. <. V ,  E >.  /  x ]. [. N  /  y ]. { y ,  n }  e.  ran  ( 2nd `  x ) } )
3 sbcel1g 3776 . . . . . 6  |-  ( N  e.  V  ->  ( [. N  /  y ]. { y ,  n }  e.  ran  ( 2nd `  x )  <->  [_ N  / 
y ]_ { y ,  n }  e.  ran  ( 2nd `  x ) ) )
43adantl 468 . . . . 5  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( [. N  /  y ]. { y ,  n }  e.  ran  ( 2nd `  x )  <->  [_ N  / 
y ]_ { y ,  n }  e.  ran  ( 2nd `  x ) ) )
54sbcbidv 3322 . . . 4  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( [. <. V ,  E >.  /  x ]. [. N  /  y ]. {
y ,  n }  e.  ran  ( 2nd `  x
)  <->  [. <. V ,  E >.  /  x ]. [_ N  /  y ]_ {
y ,  n }  e.  ran  ( 2nd `  x
) ) )
6 sbcel2 3778 . . . . 5  |-  ( [. <. V ,  E >.  /  x ]. [_ N  /  y ]_ {
y ,  n }  e.  ran  ( 2nd `  x
)  <->  [_ N  /  y ]_ { y ,  n }  e.  [_ <. V ,  E >.  /  x ]_ ran  ( 2nd `  x
) )
76a1i 11 . . . 4  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( [. <. V ,  E >.  /  x ]. [_ N  /  y ]_ {
y ,  n }  e.  ran  ( 2nd `  x
)  <->  [_ N  /  y ]_ { y ,  n }  e.  [_ <. V ,  E >.  /  x ]_ ran  ( 2nd `  x
) ) )
8 df-pr 3971 . . . . . . . 8  |-  { y ,  n }  =  ( { y }  u.  { n } )
98a1i 11 . . . . . . 7  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  { y ,  n }  =  ( { y }  u.  { n } ) )
109csbeq2dv 3781 . . . . . 6  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  [_ N  /  y ]_ {
y ,  n }  =  [_ N  /  y ]_ ( { y }  u.  { n }
) )
11 nfcv 2592 . . . . . . . . 9  |-  F/_ y
( { N }  u.  { n } )
1211a1i 11 . . . . . . . 8  |-  ( N  e.  V  ->  F/_ y
( { N }  u.  { n } ) )
13 sneq 3978 . . . . . . . . 9  |-  ( y  =  N  ->  { y }  =  { N } )
1413uneq1d 3587 . . . . . . . 8  |-  ( y  =  N  ->  ( { y }  u.  { n } )  =  ( { N }  u.  { n } ) )
1512, 14csbiegf 3387 . . . . . . 7  |-  ( N  e.  V  ->  [_ N  /  y ]_ ( { y }  u.  { n } )  =  ( { N }  u.  { n } ) )
1615adantl 468 . . . . . 6  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  [_ N  /  y ]_ ( { y }  u.  { n } )  =  ( { N }  u.  { n } ) )
17 df-pr 3971 . . . . . . . 8  |-  { N ,  n }  =  ( { N }  u.  { n } )
1817eqcomi 2460 . . . . . . 7  |-  ( { N }  u.  {
n } )  =  { N ,  n }
1918a1i 11 . . . . . 6  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( { N }  u.  {
n } )  =  { N ,  n } )
2010, 16, 193eqtrd 2489 . . . . 5  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  [_ N  /  y ]_ {
y ,  n }  =  { N ,  n } )
21 csbrn 5297 . . . . . . 7  |-  [_ <. V ,  E >.  /  x ]_ ran  ( 2nd `  x
)  =  ran  [_ <. V ,  E >.  /  x ]_ ( 2nd `  x
)
2221a1i 11 . . . . . 6  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  [_ <. V ,  E >.  /  x ]_ ran  ( 2nd `  x
)  =  ran  [_ <. V ,  E >.  /  x ]_ ( 2nd `  x
) )
23 opex 4664 . . . . . . . . 9  |-  <. V ,  E >.  e.  _V
24 csbfv2g 5901 . . . . . . . . 9  |-  ( <. V ,  E >.  e. 
_V  ->  [_ <. V ,  E >.  /  x ]_ ( 2nd `  x )  =  ( 2nd `  [_ <. V ,  E >.  /  x ]_ x ) )
2523, 24mp1i 13 . . . . . . . 8  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  [_ <. V ,  E >.  /  x ]_ ( 2nd `  x
)  =  ( 2nd `  [_ <. V ,  E >.  /  x ]_ x
) )
26 csbvarg 3792 . . . . . . . . . 10  |-  ( <. V ,  E >.  e. 
_V  ->  [_ <. V ,  E >.  /  x ]_ x  =  <. V ,  E >. )
2723, 26mp1i 13 . . . . . . . . 9  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  [_ <. V ,  E >.  /  x ]_ x  =  <. V ,  E >. )
2827fveq2d 5869 . . . . . . . 8  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( 2nd `  [_ <. V ,  E >.  /  x ]_ x )  =  ( 2nd `  <. V ,  E >. ) )
29 op2ndg 6806 . . . . . . . . 9  |-  ( ( V  e.  Y  /\  E  e.  Z )  ->  ( 2nd `  <. V ,  E >. )  =  E )
3029adantr 467 . . . . . . . 8  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( 2nd `  <. V ,  E >. )  =  E )
3125, 28, 303eqtrd 2489 . . . . . . 7  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  [_ <. V ,  E >.  /  x ]_ ( 2nd `  x
)  =  E )
3231rneqd 5062 . . . . . 6  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ran  [_
<. V ,  E >.  /  x ]_ ( 2nd `  x )  =  ran  E )
3322, 32eqtrd 2485 . . . . 5  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  [_ <. V ,  E >.  /  x ]_ ran  ( 2nd `  x
)  =  ran  E
)
3420, 33eleq12d 2523 . . . 4  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( [_ N  /  y ]_ { y ,  n }  e.  [_ <. V ,  E >.  /  x ]_ ran  ( 2nd `  x
)  <->  { N ,  n }  e.  ran  E ) )
355, 7, 343bitrd 283 . . 3  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( [. <. V ,  E >.  /  x ]. [. N  /  y ]. {
y ,  n }  e.  ran  ( 2nd `  x
)  <->  { N ,  n }  e.  ran  E ) )
3635rabbidv 3036 . 2  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  { n  e.  V  |  [. <. V ,  E >.  /  x ]. [. N  /  y ]. { y ,  n }  e.  ran  ( 2nd `  x ) }  =  { n  e.  V  |  { N ,  n }  e.  ran  E }
)
372, 36eqtrd 2485 1  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   F/_wnfc 2579   {crab 2741   _Vcvv 3045   [.wsbc 3267   [_csb 3363    u. cun 3402   {csn 3968   {cpr 3970   <.cop 3974   ran crn 4835   ` cfv 5582  (class class class)co 6290   2ndc2nd 6792   Neighbors cnbgra 25145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-nbgra 25148
This theorem is referenced by: (None)
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