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Theorem nbgraf1olem3 25164
Description: Lemma 3 for nbgraf1o 25168. 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 2520 . . . 4  |-  ( M  e.  N  <->  M  e.  ( <. V ,  E >. Neighbors  U ) )
3 nbgracnvfv 25161 . . . 4  |-  ( ( V USGrph  E  /\  M  e.  ( <. V ,  E >. Neighbors  U ) )  -> 
( E `  ( `' E `  { U ,  M } ) )  =  { U ,  M } )
42, 3sylan2b 478 . . 3  |-  ( ( V USGrph  E  /\  M  e.  N )  ->  ( E `  ( `' E `  { U ,  M } ) )  =  { U ,  M } )
543adant2 1026 . 2  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  ( E `  ( `' E `  { U ,  M } ) )  =  { U ,  M } )
6 usgraf1o 25078 . . . . . . 7  |-  ( V USGrph  E  ->  E : dom  E -1-1-onto-> ran 
E )
763ad2ant1 1028 . . . . . 6  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  E : dom  E -1-1-onto-> ran  E )
8 nbgraeledg 25151 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( M  e.  ( <. V ,  E >. Neighbors  U )  <->  { M ,  U }  e.  ran  E ) )
9 prcom 4049 . . . . . . . . . . . . 13  |-  { M ,  U }  =  { U ,  M }
109eleq1i 2519 . . . . . . . . . . . 12  |-  ( { M ,  U }  e.  ran  E  <->  { U ,  M }  e.  ran  E )
118, 10syl6bb 265 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( M  e.  ( <. V ,  E >. Neighbors  U )  <->  { U ,  M }  e.  ran  E ) )
1211biimpcd 228 . . . . . . . . . 10  |-  ( M  e.  ( <. V ,  E >. Neighbors  U )  ->  ( V USGrph  E  ->  { U ,  M }  e.  ran  E ) )
1312a1d 26 . . . . . . . . 9  |-  ( M  e.  ( <. V ,  E >. Neighbors  U )  ->  ( U  e.  V  ->  ( V USGrph  E  ->  { U ,  M }  e.  ran  E ) ) )
1413, 1eleq2s 2546 . . . . . . . 8  |-  ( M  e.  N  ->  ( U  e.  V  ->  ( V USGrph  E  ->  { U ,  M }  e.  ran  E ) ) )
1514com13 83 . . . . . . 7  |-  ( V USGrph  E  ->  ( U  e.  V  ->  ( M  e.  N  ->  { U ,  M }  e.  ran  E ) ) )
16153imp 1201 . . . . . 6  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  { U ,  M }  e.  ran  E )
17 f1ocnvdm 6181 . . . . . 6  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { U ,  M }  e.  ran  E )  ->  ( `' E `  { U ,  M } )  e. 
dom  E )
187, 16, 17syl2anc 666 . . . . 5  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  ( `' E `  { U ,  M } )  e. 
dom  E )
19 prid1g 4077 . . . . . . . 8  |-  ( U  e.  V  ->  U  e.  { U ,  M } )
20 eleq2 2517 . . . . . . . 8  |-  ( ( E `  ( `' E `  { U ,  M } ) )  =  { U ,  M }  ->  ( U  e.  ( E `  ( `' E `  { U ,  M } ) )  <-> 
U  e.  { U ,  M } ) )
2119, 20syl5ibrcom 226 . . . . . . 7  |-  ( U  e.  V  ->  (
( E `  ( `' E `  { U ,  M } ) )  =  { U ,  M }  ->  U  e.  ( E `  ( `' E `  { U ,  M } ) ) ) )
22213ad2ant2 1029 . . . . . 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 5863 . . . . . . 7  |-  ( i  =  ( `' E `  { U ,  M } )  ->  ( E `  i )  =  ( E `  ( `' E `  { U ,  M } ) ) )
2524eleq2d 2513 . . . . . 6  |-  ( i  =  ( `' E `  { U ,  M } )  ->  ( U  e.  ( E `  i )  <->  U  e.  ( E `  ( `' E `  { U ,  M } ) ) ) )
2625elrab 3195 . . . . 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 669 . . . 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 2539 . . 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 25162 . . . 4  |-  ( ( ( V USGrph  E  /\  U  e.  V )  /\  M  e.  N
)  ->  E! i  e.  I  ( E `  i )  =  { U ,  M }
)
32313impa 1202 . . 3  |-  ( ( V USGrph  E  /\  U  e.  V  /\  M  e.  N )  ->  E! i  e.  I  ( E `  i )  =  { U ,  M } )
3324eqeq1d 2452 . . . 4  |-  ( i  =  ( `' E `  { U ,  M } )  ->  (
( E `  i
)  =  { U ,  M }  <->  ( E `  ( `' E `  { U ,  M }
) )  =  { U ,  M }
) )
3433riota2 6272 . . 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 666 . 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 214 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 188    /\ w3a 984    = wceq 1443    e. wcel 1886   E!wreu 2738   {crab 2740   {cpr 3969   <.cop 3973   class class class wbr 4401    |-> cmpt 4460   `'ccnv 4832   dom cdm 4833   ran crn 4834   -1-1-onto->wf1o 5580   ` cfv 5581   iota_crio 6249  (class class class)co 6288   USGrph cusg 25050   Neighbors cnbgra 25138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-hash 12513  df-usgra 25053  df-nbgra 25141
This theorem is referenced by:  nbgraf1olem4  25165  nbgraf1olem5  25166
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