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Theorem nbgraf1olem3 24648
Description: Lemma 3 for nbgraf1o 24652. 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 24645 . . . 4  |-  ( ( V USGrph  E  /\  M  e.  ( <. V ,  E >. Neighbors  U ) )  -> 
( E `  ( `' E `  { U ,  M } ) )  =  { U ,  M } )
42, 3sylan2b 473 . . 3  |-  ( ( V USGrph  E  /\  M  e.  N )  ->  ( E `  ( `' E `  { U ,  M } ) )  =  { U ,  M } )
543adant2 1013 . 2  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  ( E `  ( `' E `  { U ,  M } ) )  =  { U ,  M } )
6 usgraf1o 24563 . . . . . . 7  |-  ( V USGrph  E  ->  E : dom  E -1-1-onto-> ran 
E )
763ad2ant1 1015 . . . . . 6  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  E : dom  E -1-1-onto-> ran  E )
8 nbgraeledg 24635 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( M  e.  ( <. V ,  E >. Neighbors  U )  <->  { M ,  U }  e.  ran  E ) )
9 prcom 4094 . . . . . . . . . . . . 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 1188 . . . . . 6  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  { U ,  M }  e.  ran  E )
17 f1ocnvdm 6163 . . . . . 6  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { U ,  M }  e.  ran  E )  ->  ( `' E `  { U ,  M } )  e. 
dom  E )
187, 16, 17syl2anc 659 . . . . 5  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  ( `' E `  { U ,  M } )  e. 
dom  E )
19 prid1g 4122 . . . . . . . 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 1016 . . . . . 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 5848 . . . . . . 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 3254 . . . . 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 662 . . . 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 24646 . . . 4  |-  ( ( ( V USGrph  E  /\  U  e.  V )  /\  M  e.  N
)  ->  E! i  e.  I  ( E `  i )  =  { U ,  M }
)
32313impa 1189 . . 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 6254 . . 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 659 . 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 971    = wceq 1398    e. wcel 1823   E!wreu 2806   {crab 2808   {cpr 4018   <.cop 4022   class class class wbr 4439    |-> cmpt 4497   `'ccnv 4987   dom cdm 4988   ran crn 4989   -1-1-onto->wf1o 5569   ` cfv 5570   iota_crio 6231  (class class class)co 6270   USGrph cusg 24535   Neighbors cnbgra 24622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-hash 12391  df-usgra 24538  df-nbgra 24625
This theorem is referenced by:  nbgraf1olem4  24649  nbgraf1olem5  24650
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