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Theorem nbgraf1olem3 23505
Description: Lemma 3 for nbgraf1o 23509. 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 2532 . . . 4  |-  ( M  e.  N  <->  M  e.  ( <. V ,  E >. Neighbors  U ) )
3 nbgracnvfv 23502 . . . 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 1007 . 2  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  ( E `  ( `' E `  { U ,  M } ) )  =  { U ,  M } )
6 usgraf1o 23434 . . . . . . 7  |-  ( V USGrph  E  ->  E : dom  E -1-1-onto-> ran 
E )
763ad2ant1 1009 . . . . . 6  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  E : dom  E -1-1-onto-> ran  E )
8 nbgraeledg 23494 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( M  e.  ( <. V ,  E >. Neighbors  U )  <->  { M ,  U }  e.  ran  E ) )
9 prcom 4062 . . . . . . . . . . . . 13  |-  { M ,  U }  =  { U ,  M }
109eleq1i 2531 . . . . . . . . . . . 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 2562 . . . . . . . 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 1182 . . . . . 6  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  { U ,  M }  e.  ran  E )
17 f1ocnvdm 6099 . . . . . 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 4090 . . . . . . . 8  |-  ( U  e.  V  ->  U  e.  { U ,  M } )
20 eleq2 2527 . . . . . . . 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 1010 . . . . . 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 5800 . . . . . . 7  |-  ( i  =  ( `' E `  { U ,  M } )  ->  ( E `  i )  =  ( E `  ( `' E `  { U ,  M } ) ) )
2524eleq2d 2524 . . . . . 6  |-  ( i  =  ( `' E `  { U ,  M } )  ->  ( U  e.  ( E `  i )  <->  U  e.  ( E `  ( `' E `  { U ,  M } ) ) ) )
2625elrab 3224 . . . . 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 2553 . . 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 23503 . . . 4  |-  ( ( ( V USGrph  E  /\  U  e.  V )  /\  M  e.  N
)  ->  E! i  e.  I  ( E `  i )  =  { U ,  M }
)
32313impa 1183 . . 3  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  E! i  e.  I  ( E `  i )  =  { U ,  M } )
3324eqeq1d 2456 . . . 4  |-  ( i  =  ( `' E `  { U ,  M } )  ->  (
( E `  i
)  =  { U ,  M }  <->  ( E `  ( `' E `  { U ,  M }
) )  =  { U ,  M }
) )
3433riota2 6185 . . 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 965    = wceq 1370    e. wcel 1758   E!wreu 2801   {crab 2803   {cpr 3988   <.cop 3992   class class class wbr 4401    |-> cmpt 4459   `'ccnv 4948   dom cdm 4949   ran crn 4950   -1-1-onto->wf1o 5526   ` cfv 5527   iota_crio 6161  (class class class)co 6201   USGrph cusg 23417   Neighbors cnbgra 23482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-hash 12222  df-usgra 23419  df-nbgra 23485
This theorem is referenced by:  nbgraf1olem4  23506  nbgraf1olem5  23507
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