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Theorem frgrancvvdeqlemB 24703
Description: Lemma B for frgrancvvdeq 24707. This corresponds to the following observation in [Huneke] p. 1: "The map is one-to-one since z in N(x) is uniquely determined as the common neighbor of x and a(x)". (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
frgrancvvdeqlemB  |-  ( ph  ->  A : D -1-1-> ran  A )
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 frgrancvvdeqlemB
Dummy variables  w  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrancvvdeq.nx . . 3  |-  D  =  ( <. V ,  E >. Neighbors  X )
2 frgrancvvdeq.ny . . 3  |-  N  =  ( <. V ,  E >. Neighbors  Y )
3 frgrancvvdeq.x . . 3  |-  ( ph  ->  X  e.  V )
4 frgrancvvdeq.y . . 3  |-  ( ph  ->  Y  e.  V )
5 frgrancvvdeq.ne . . 3  |-  ( ph  ->  X  =/=  Y )
6 frgrancvvdeq.xy . . 3  |-  ( ph  ->  Y  e/  D )
7 frgrancvvdeq.f . . 3  |-  ( ph  ->  V FriendGrph  E )
8 frgrancvvdeq.a . . 3  |-  A  =  ( x  e.  D  |->  ( iota_ y  e.  N  { x ,  y }  e.  ran  E
) )
91, 2, 3, 4, 5, 6, 7, 8frgrancvvdeqlem5 24699 . 2  |-  ( ph  ->  A : D --> N )
10 ffn 5724 . . . . . 6  |-  ( A : D --> N  ->  A  Fn  D )
11 dffn3 5731 . . . . . 6  |-  ( A  Fn  D  <->  A : D
--> ran  A )
1210, 11sylib 196 . . . . 5  |-  ( A : D --> N  ->  A : D --> ran  A
)
1312adantl 466 . . . 4  |-  ( (
ph  /\  A : D
--> N )  ->  A : D --> ran  A )
14 ffvelrn 6012 . . . . . . . . . . . 12  |-  ( ( A : D --> N  /\  u  e.  D )  ->  ( A `  u
)  e.  N )
1514adantll 713 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A : D --> N )  /\  u  e.  D )  ->  ( A `  u
)  e.  N )
1615adantr 465 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  A : D --> N )  /\  u  e.  D
)  /\  w  e.  D )  ->  ( A `  u )  e.  N )
1716adantr 465 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  A : D --> N )  /\  u  e.  D
)  /\  w  e.  D )  /\  ( A `  u )  =  ( A `  w ) )  -> 
( A `  u
)  e.  N )
181, 2, 3, 4, 5, 6, 7, 8frgrancvvdeqlem2 24696 . . . . . . . . . . . . 13  |-  ( ph  ->  X  e/  N )
19 preq1 4101 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  =  u  ->  { x ,  y }  =  { u ,  y } )
2019eleq1d 2531 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  =  u  ->  ( { x ,  y }  e.  ran  E  <->  { u ,  y }  e.  ran  E ) )
2120riotabidv 6240 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  =  u  ->  ( iota_ y  e.  N  {
x ,  y }  e.  ran  E )  =  ( iota_ y  e.  N  { u ,  y }  e.  ran  E ) )
2221cbvmptv 4533 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  D  |->  ( iota_ y  e.  N  { x ,  y }  e.  ran  E ) )  =  ( u  e.  D  |->  ( iota_ y  e.  N  { u ,  y }  e.  ran  E
) )
238, 22eqtri 2491 . . . . . . . . . . . . . . . . . . 19  |-  A  =  ( u  e.  D  |->  ( iota_ y  e.  N  { u ,  y }  e.  ran  E
) )
241, 2, 3, 4, 5, 6, 7, 23frgrancvvdeqlem7 24701 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  u  e.  D )  ->  { u ,  ( A `  u ) }  e.  ran  E )
25 preq1 4101 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  =  w  ->  { x ,  y }  =  { w ,  y } )
2625eleq1d 2531 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  =  w  ->  ( { x ,  y }  e.  ran  E  <->  { w ,  y }  e.  ran  E ) )
2726riotabidv 6240 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  =  w  ->  ( iota_ y  e.  N  {
x ,  y }  e.  ran  E )  =  ( iota_ y  e.  N  { w ,  y }  e.  ran  E ) )
2827cbvmptv 4533 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  D  |->  ( iota_ y  e.  N  { x ,  y }  e.  ran  E ) )  =  ( w  e.  D  |->  ( iota_ y  e.  N  { w ,  y }  e.  ran  E
) )
298, 28eqtri 2491 . . . . . . . . . . . . . . . . . . 19  |-  A  =  ( w  e.  D  |->  ( iota_ y  e.  N  { w ,  y }  e.  ran  E
) )
301, 2, 3, 4, 5, 6, 7, 29frgrancvvdeqlem7 24701 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  w  e.  D )  ->  { w ,  ( A `  w ) }  e.  ran  E )
3124, 30anim12dan 834 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( u  e.  D  /\  w  e.  D ) )  -> 
( { u ,  ( A `  u
) }  e.  ran  E  /\  { w ,  ( A `  w
) }  e.  ran  E ) )
32 preq2 4102 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( A `  w )  =  ( A `  u )  ->  { w ,  ( A `  w ) }  =  { w ,  ( A `  u ) } )
3332eleq1d 2531 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( A `  w )  =  ( A `  u )  ->  ( { w ,  ( A `  w ) }  e.  ran  E  <->  { w ,  ( A `
 u ) }  e.  ran  E ) )
3433anbi2d 703 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( A `  w )  =  ( A `  u )  ->  (
( { u ,  ( A `  u
) }  e.  ran  E  /\  { w ,  ( A `  w
) }  e.  ran  E )  <->  ( { u ,  ( A `  u ) }  e.  ran  E  /\  { w ,  ( A `  u ) }  e.  ran  E ) ) )
3534eqcoms 2474 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A `  u )  =  ( A `  w )  ->  (
( { u ,  ( A `  u
) }  e.  ran  E  /\  { w ,  ( A `  w
) }  e.  ran  E )  <->  ( { u ,  ( A `  u ) }  e.  ran  E  /\  { w ,  ( A `  u ) }  e.  ran  E ) ) )
3635biimpa 484 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( A `  u
)  =  ( A `
 w )  /\  ( { u ,  ( A `  u ) }  e.  ran  E  /\  { w ,  ( A `  w ) }  e.  ran  E
) )  ->  ( { u ,  ( A `  u ) }  e.  ran  E  /\  { w ,  ( A `  u ) }  e.  ran  E
) )
37 df-ne 2659 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =/=  w  <->  -.  u  =  w )
383, 1, 7frgranbnb 24684 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
ph  /\  ( u  e.  D  /\  w  e.  D )  /\  u  =/=  w )  ->  (
( { u ,  ( A `  u
) }  e.  ran  E  /\  { w ,  ( A `  u
) }  e.  ran  E )  ->  ( A `  u )  =  X ) )
39383expa 1191 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ph  /\  (
u  e.  D  /\  w  e.  D )
)  /\  u  =/=  w )  ->  (
( { u ,  ( A `  u
) }  e.  ran  E  /\  { w ,  ( A `  u
) }  e.  ran  E )  ->  ( A `  u )  =  X ) )
40 df-nel 2660 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( X  e/  N  <->  -.  X  e.  N )
41 eleq1 2534 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( A `  u )  =  X  ->  (
( A `  u
)  e.  N  <->  X  e.  N ) )
4241biimpa 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( A `  u
)  =  X  /\  ( A `  u )  e.  N )  ->  X  e.  N )
4342pm2.24d 143 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( A `  u
)  =  X  /\  ( A `  u )  e.  N )  -> 
( -.  X  e.  N  ->  u  =  w ) )
4443expcom 435 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( A `  u )  e.  N  ->  (
( A `  u
)  =  X  -> 
( -.  X  e.  N  ->  u  =  w ) ) )
4544com13 80 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( -.  X  e.  N  -> 
( ( A `  u )  =  X  ->  ( ( A `
 u )  e.  N  ->  u  =  w ) ) )
4640, 45sylbi 195 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( X  e/  N  ->  (
( A `  u
)  =  X  -> 
( ( A `  u )  e.  N  ->  u  =  w ) ) )
4746com12 31 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( A `  u )  =  X  ->  ( X  e/  N  ->  (
( A `  u
)  e.  N  ->  u  =  w )
) )
4839, 47syl6 33 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  (
u  e.  D  /\  w  e.  D )
)  /\  u  =/=  w )  ->  (
( { u ,  ( A `  u
) }  e.  ran  E  /\  { w ,  ( A `  u
) }  e.  ran  E )  ->  ( X  e/  N  ->  ( ( A `  u )  e.  N  ->  u  =  w ) ) ) )
4948expcom 435 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =/=  w  ->  (
( ph  /\  (
u  e.  D  /\  w  e.  D )
)  ->  ( ( { u ,  ( A `  u ) }  e.  ran  E  /\  { w ,  ( A `  u ) }  e.  ran  E
)  ->  ( X  e/  N  ->  ( ( A `  u )  e.  N  ->  u  =  w ) ) ) ) )
5049com23 78 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  =/=  w  ->  (
( { u ,  ( A `  u
) }  e.  ran  E  /\  { w ,  ( A `  u
) }  e.  ran  E )  ->  ( ( ph  /\  ( u  e.  D  /\  w  e.  D ) )  -> 
( X  e/  N  ->  ( ( A `  u )  e.  N  ->  u  =  w ) ) ) ) )
5137, 50sylbir 213 . . . . . . . . . . . . . . . . . . . 20  |-  ( -.  u  =  w  -> 
( ( { u ,  ( A `  u ) }  e.  ran  E  /\  { w ,  ( A `  u ) }  e.  ran  E )  ->  (
( ph  /\  (
u  e.  D  /\  w  e.  D )
)  ->  ( X  e/  N  ->  ( ( A `  u )  e.  N  ->  u  =  w ) ) ) ) )
5236, 51syl5com 30 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A `  u
)  =  ( A `
 w )  /\  ( { u ,  ( A `  u ) }  e.  ran  E  /\  { w ,  ( A `  w ) }  e.  ran  E
) )  ->  ( -.  u  =  w  ->  ( ( ph  /\  ( u  e.  D  /\  w  e.  D
) )  ->  ( X  e/  N  ->  (
( A `  u
)  e.  N  ->  u  =  w )
) ) ) )
5352expcom 435 . . . . . . . . . . . . . . . . . 18  |-  ( ( { u ,  ( A `  u ) }  e.  ran  E  /\  { w ,  ( A `  w ) }  e.  ran  E
)  ->  ( ( A `  u )  =  ( A `  w )  ->  ( -.  u  =  w  ->  ( ( ph  /\  ( u  e.  D  /\  w  e.  D
) )  ->  ( X  e/  N  ->  (
( A `  u
)  e.  N  ->  u  =  w )
) ) ) ) )
5453com24 87 . . . . . . . . . . . . . . . . 17  |-  ( ( { u ,  ( A `  u ) }  e.  ran  E  /\  { w ,  ( A `  w ) }  e.  ran  E
)  ->  ( ( ph  /\  ( u  e.  D  /\  w  e.  D ) )  -> 
( -.  u  =  w  ->  ( ( A `  u )  =  ( A `  w )  ->  ( X  e/  N  ->  (
( A `  u
)  e.  N  ->  u  =  w )
) ) ) ) )
5531, 54mpcom 36 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( u  e.  D  /\  w  e.  D ) )  -> 
( -.  u  =  w  ->  ( ( A `  u )  =  ( A `  w )  ->  ( X  e/  N  ->  (
( A `  u
)  e.  N  ->  u  =  w )
) ) ) )
5655ex 434 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( u  e.  D  /\  w  e.  D )  ->  ( -.  u  =  w  ->  ( ( A `  u )  =  ( A `  w )  ->  ( X  e/  N  ->  ( ( A `
 u )  e.  N  ->  u  =  w ) ) ) ) ) )
5756com3r 79 . . . . . . . . . . . . . 14  |-  ( -.  u  =  w  -> 
( ph  ->  ( ( u  e.  D  /\  w  e.  D )  ->  ( ( A `  u )  =  ( A `  w )  ->  ( X  e/  N  ->  ( ( A `
 u )  e.  N  ->  u  =  w ) ) ) ) ) )
5857com15 93 . . . . . . . . . . . . 13  |-  ( X  e/  N  ->  ( ph  ->  ( ( u  e.  D  /\  w  e.  D )  ->  (
( A `  u
)  =  ( A `
 w )  -> 
( -.  u  =  w  ->  ( ( A `  u )  e.  N  ->  u  =  w ) ) ) ) ) )
5918, 58mpcom 36 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( u  e.  D  /\  w  e.  D )  ->  (
( A `  u
)  =  ( A `
 w )  -> 
( -.  u  =  w  ->  ( ( A `  u )  e.  N  ->  u  =  w ) ) ) ) )
6059expd 436 . . . . . . . . . . 11  |-  ( ph  ->  ( u  e.  D  ->  ( w  e.  D  ->  ( ( A `  u )  =  ( A `  w )  ->  ( -.  u  =  w  ->  ( ( A `  u )  e.  N  ->  u  =  w ) ) ) ) ) )
6160adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  A : D
--> N )  ->  (
u  e.  D  -> 
( w  e.  D  ->  ( ( A `  u )  =  ( A `  w )  ->  ( -.  u  =  w  ->  ( ( A `  u )  e.  N  ->  u  =  w ) ) ) ) ) )
6261imp41 593 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  A : D --> N )  /\  u  e.  D
)  /\  w  e.  D )  /\  ( A `  u )  =  ( A `  w ) )  -> 
( -.  u  =  w  ->  ( ( A `  u )  e.  N  ->  u  =  w ) ) )
6317, 62mpid 41 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  A : D --> N )  /\  u  e.  D
)  /\  w  e.  D )  /\  ( A `  u )  =  ( A `  w ) )  -> 
( -.  u  =  w  ->  u  =  w ) )
6463pm2.18d 111 . . . . . . 7  |-  ( ( ( ( ( ph  /\  A : D --> N )  /\  u  e.  D
)  /\  w  e.  D )  /\  ( A `  u )  =  ( A `  w ) )  ->  u  =  w )
6564ex 434 . . . . . 6  |-  ( ( ( ( ph  /\  A : D --> N )  /\  u  e.  D
)  /\  w  e.  D )  ->  (
( A `  u
)  =  ( A `
 w )  ->  u  =  w )
)
6665ralrimiva 2873 . . . . 5  |-  ( ( ( ph  /\  A : D --> N )  /\  u  e.  D )  ->  A. w  e.  D  ( ( A `  u )  =  ( A `  w )  ->  u  =  w ) )
6766ralrimiva 2873 . . . 4  |-  ( (
ph  /\  A : D
--> N )  ->  A. u  e.  D  A. w  e.  D  ( ( A `  u )  =  ( A `  w )  ->  u  =  w ) )
68 dff13 6147 . . . 4  |-  ( A : D -1-1-> ran  A  <->  ( A : D --> ran  A  /\  A. u  e.  D  A. w  e.  D  ( ( A `  u )  =  ( A `  w )  ->  u  =  w ) ) )
6913, 67, 68sylanbrc 664 . . 3  |-  ( (
ph  /\  A : D
--> N )  ->  A : D -1-1-> ran  A )
7069expcom 435 . 2  |-  ( A : D --> N  -> 
( ph  ->  A : D -1-1-> ran  A ) )
719, 70mpcom 36 1  |-  ( ph  ->  A : D -1-1-> ran  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2657    e/ wnel 2658   A.wral 2809   {cpr 4024   <.cop 4028   class class class wbr 4442    |-> cmpt 4500   ran crn 4995    Fn wfn 5576   -->wf 5577   -1-1->wf1 5578   ` cfv 5581   iota_crio 6237  (class class class)co 6277   Neighbors cnbgra 24081   FriendGrph cfrgra 24652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-hash 12363  df-usgra 23998  df-nbgra 24084  df-frgra 24653
This theorem is referenced by:  frgrancvvdeqlem8  24705
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