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Theorem nbgraf1olem3 24116
Description: Lemma 3 for nbgraf1o 24120. The restricted iota of an edge is the function value of the converse applied to the edge. (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
nbgraf1olem3  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  ( iota_ i  e.  I  ( E `  i )  =  { U ,  M } )  =  ( `' E `  { U ,  M } ) )
Distinct variable groups:    i, E, n    U, i, n    i, V, n    i, I, n   
n, N    i, M
Allowed substitution hints:    F( i, n)    M( n)    N( i)

Proof of Theorem nbgraf1olem3
StepHypRef Expression
1 nbgraf1o.n . . . . 5  |-  N  =  ( <. V ,  E >. Neighbors  U )
21eleq2i 2545 . . . 4  |-  ( M  e.  N  <->  M  e.  ( <. V ,  E >. Neighbors  U ) )
3 nbgracnvfv 24113 . . . 4  |-  ( ( V USGrph  E  /\  M  e.  ( <. V ,  E >. Neighbors  U ) )  -> 
( E `  ( `' E `  { U ,  M } ) )  =  { U ,  M } )
42, 3sylan2b 475 . . 3  |-  ( ( V USGrph  E  /\  M  e.  N )  ->  ( E `  ( `' E `  { U ,  M } ) )  =  { U ,  M } )
543adant2 1015 . 2  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  ( E `  ( `' E `  { U ,  M } ) )  =  { U ,  M } )
6 usgraf1o 24031 . . . . . . 7  |-  ( V USGrph  E  ->  E : dom  E -1-1-onto-> ran 
E )
763ad2ant1 1017 . . . . . 6  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  E : dom  E -1-1-onto-> ran  E )
8 nbgraeledg 24103 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( M  e.  ( <. V ,  E >. Neighbors  U )  <->  { M ,  U }  e.  ran  E ) )
9 prcom 4105 . . . . . . . . . . . . 13  |-  { M ,  U }  =  { U ,  M }
109eleq1i 2544 . . . . . . . . . . . 12  |-  ( { M ,  U }  e.  ran  E  <->  { U ,  M }  e.  ran  E )
118, 10syl6bb 261 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( M  e.  ( <. V ,  E >. Neighbors  U )  <->  { U ,  M }  e.  ran  E ) )
1211biimpcd 224 . . . . . . . . . 10  |-  ( M  e.  ( <. V ,  E >. Neighbors  U )  ->  ( V USGrph  E  ->  { U ,  M }  e.  ran  E ) )
1312a1d 25 . . . . . . . . 9  |-  ( M  e.  ( <. V ,  E >. Neighbors  U )  ->  ( U  e.  V  ->  ( V USGrph  E  ->  { U ,  M }  e.  ran  E ) ) )
1413, 1eleq2s 2575 . . . . . . . 8  |-  ( M  e.  N  ->  ( U  e.  V  ->  ( V USGrph  E  ->  { U ,  M }  e.  ran  E ) ) )
1514com13 80 . . . . . . 7  |-  ( V USGrph  E  ->  ( U  e.  V  ->  ( M  e.  N  ->  { U ,  M }  e.  ran  E ) ) )
16153imp 1190 . . . . . 6  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  { U ,  M }  e.  ran  E )
17 f1ocnvdm 6174 . . . . . 6  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { U ,  M }  e.  ran  E )  ->  ( `' E `  { U ,  M } )  e. 
dom  E )
187, 16, 17syl2anc 661 . . . . 5  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  ( `' E `  { U ,  M } )  e. 
dom  E )
19 prid1g 4133 . . . . . . . 8  |-  ( U  e.  V  ->  U  e.  { U ,  M } )
20 eleq2 2540 . . . . . . . 8  |-  ( ( E `  ( `' E `  { U ,  M } ) )  =  { U ,  M }  ->  ( U  e.  ( E `  ( `' E `  { U ,  M } ) )  <-> 
U  e.  { U ,  M } ) )
2119, 20syl5ibrcom 222 . . . . . . 7  |-  ( U  e.  V  ->  (
( E `  ( `' E `  { U ,  M } ) )  =  { U ,  M }  ->  U  e.  ( E `  ( `' E `  { U ,  M } ) ) ) )
22213ad2ant2 1018 . . . . . 6  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  (
( E `  ( `' E `  { U ,  M } ) )  =  { U ,  M }  ->  U  e.  ( E `  ( `' E `  { U ,  M } ) ) ) )
235, 22mpd 15 . . . . 5  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  U  e.  ( E `  ( `' E `  { U ,  M } ) ) )
24 fveq2 5864 . . . . . . 7  |-  ( i  =  ( `' E `  { U ,  M } )  ->  ( E `  i )  =  ( E `  ( `' E `  { U ,  M } ) ) )
2524eleq2d 2537 . . . . . 6  |-  ( i  =  ( `' E `  { U ,  M } )  ->  ( U  e.  ( E `  i )  <->  U  e.  ( E `  ( `' E `  { U ,  M } ) ) ) )
2625elrab 3261 . . . . 5  |-  ( ( `' E `  { U ,  M } )  e. 
{ i  e.  dom  E  |  U  e.  ( E `  i ) }  <->  ( ( `' E `  { U ,  M } )  e. 
dom  E  /\  U  e.  ( E `  ( `' E `  { U ,  M } ) ) ) )
2718, 23, 26sylanbrc 664 . . . 4  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  ( `' E `  { U ,  M } )  e. 
{ i  e.  dom  E  |  U  e.  ( E `  i ) } )
28 nbgraf1o.i . . . 4  |-  I  =  { i  e.  dom  E  |  U  e.  ( E `  i ) }
2927, 28syl6eleqr 2566 . . 3  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  ( `' E `  { U ,  M } )  e.  I )
30 nbgraf1o.f . . . . 5  |-  F  =  ( n  e.  N  |->  ( iota_ i  e.  I 
( E `  i
)  =  { U ,  n } ) )
311, 28, 30nbgraf1olem1 24114 . . . 4  |-  ( ( ( V USGrph  E  /\  U  e.  V )  /\  M  e.  N
)  ->  E! i  e.  I  ( E `  i )  =  { U ,  M }
)
32313impa 1191 . . 3  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  E! i  e.  I  ( E `  i )  =  { U ,  M } )
3324eqeq1d 2469 . . . 4  |-  ( i  =  ( `' E `  { U ,  M } )  ->  (
( E `  i
)  =  { U ,  M }  <->  ( E `  ( `' E `  { U ,  M }
) )  =  { U ,  M }
) )
3433riota2 6266 . . 3  |-  ( ( ( `' E `  { U ,  M }
)  e.  I  /\  E! i  e.  I 
( E `  i
)  =  { U ,  M } )  -> 
( ( E `  ( `' E `  { U ,  M } ) )  =  { U ,  M }  <->  ( iota_ i  e.  I  ( E `  i )  =  { U ,  M }
)  =  ( `' E `  { U ,  M } ) ) )
3529, 32, 34syl2anc 661 . 2  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  (
( E `  ( `' E `  { U ,  M } ) )  =  { U ,  M }  <->  ( iota_ i  e.  I  ( E `  i )  =  { U ,  M }
)  =  ( `' E `  { U ,  M } ) ) )
365, 35mpbid 210 1  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  ( iota_ i  e.  I  ( E `  i )  =  { U ,  M } )  =  ( `' E `  { U ,  M } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767   E!wreu 2816   {crab 2818   {cpr 4029   <.cop 4033   class class class wbr 4447    |-> cmpt 4505   `'ccnv 4998   dom cdm 4999   ran crn 5000   -1-1-onto->wf1o 5585   ` cfv 5586   iota_crio 6242  (class class class)co 6282   USGrph cusg 24003   Neighbors cnbgra 24090
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 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-hash 12368  df-usgra 24006  df-nbgra 24093
This theorem is referenced by:  nbgraf1olem4  24117  nbgraf1olem5  24118
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