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Theorem nbgraf1olem4 24108
Description: Lemma 4 for nbgraf1o 24111. The mapping of neighbors to edge indices applied on a neighbor is the function value of the converse applied on the edge between the vertex and this neighbor. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
Hypotheses
Ref Expression
nbgraf1o.n  |-  N  =  ( <. V ,  E >. Neighbors  U )
nbgraf1o.i  |-  I  =  { i  e.  dom  E  |  U  e.  ( E `  i ) }
nbgraf1o.f  |-  F  =  ( n  e.  N  |->  ( iota_ i  e.  I 
( E `  i
)  =  { U ,  n } ) )
Assertion
Ref Expression
nbgraf1olem4  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  ( F `  M )  =  ( `' E `  { U ,  M } ) )
Distinct variable groups:    i, E, n    U, i, n    i, V, n    i, I, n   
n, N    i, M, n
Allowed substitution hints:    F( i, n)    N( i)

Proof of Theorem nbgraf1olem4
StepHypRef Expression
1 simp3 993 . . 3  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  M  e.  N )
2 riotaex 6242 . . 3  |-  ( iota_ i  e.  I  ( E `
 i )  =  { U ,  M } )  e.  _V
3 preq2 4102 . . . . . 6  |-  ( n  =  M  ->  { U ,  n }  =  { U ,  M }
)
43eqeq2d 2476 . . . . 5  |-  ( n  =  M  ->  (
( E `  i
)  =  { U ,  n }  <->  ( E `  i )  =  { U ,  M }
) )
54riotabidv 6240 . . . 4  |-  ( n  =  M  ->  ( iota_ i  e.  I  ( E `  i )  =  { U ,  n } )  =  (
iota_ i  e.  I 
( E `  i
)  =  { U ,  M } ) )
6 nbgraf1o.f . . . 4  |-  F  =  ( n  e.  N  |->  ( iota_ i  e.  I 
( E `  i
)  =  { U ,  n } ) )
75, 6fvmptg 5941 . . 3  |-  ( ( M  e.  N  /\  ( iota_ i  e.  I 
( E `  i
)  =  { U ,  M } )  e. 
_V )  ->  ( F `  M )  =  ( iota_ i  e.  I  ( E `  i )  =  { U ,  M }
) )
81, 2, 7sylancl 662 . 2  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  ( F `  M )  =  ( iota_ i  e.  I  ( E `  i )  =  { U ,  M }
) )
9 nbgraf1o.n . . 3  |-  N  =  ( <. V ,  E >. Neighbors  U )
10 nbgraf1o.i . . 3  |-  I  =  { i  e.  dom  E  |  U  e.  ( E `  i ) }
119, 10, 6nbgraf1olem3 24107 . 2  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  ( iota_ i  e.  I  ( E `  i )  =  { U ,  M } )  =  ( `' E `  { U ,  M } ) )
128, 11eqtrd 2503 1  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  ( F `  M )  =  ( `' E `  { U ,  M } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 968    = wceq 1374    e. wcel 1762   {crab 2813   _Vcvv 3108   {cpr 4024   <.cop 4028   class class class wbr 4442    |-> cmpt 4500   `'ccnv 4993   dom cdm 4994   ` cfv 5581   iota_crio 6237  (class class class)co 6277   USGrph cusg 23995   Neighbors cnbgra 24081
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
This theorem is referenced by:  nbgraf1olem5  24109
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