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Theorem edgnbusgreu 39605
Description: For each edge incident to a vertex there is exactly one neighbor of the vertex also incident to this edge in a simple graph. (Contributed by AV, 28-Oct-2020.)
Hypotheses
Ref Expression
edgnbusgreu.v  |-  V  =  (Vtx `  G )
edgnbusgreu.e  |-  E  =  (Edg `  G )
edgnbusgreu.n  |-  N  =  ( G NeighbVtx  M )
Assertion
Ref Expression
edgnbusgreu  |-  ( ( ( G  e. USGraph  /\  M  e.  V )  /\  ( C  e.  E  /\  M  e.  C )
)  ->  E! n  e.  N  C  =  { M ,  n }
)
Distinct variable groups:    C, n    n, E    n, G    n, M    n, V
Allowed substitution hint:    N( n)

Proof of Theorem edgnbusgreu
StepHypRef Expression
1 simpl 464 . . . . . 6  |-  ( ( G  e. USGraph  /\  M  e.  V )  ->  G  e. USGraph  )
21adantr 472 . . . . 5  |-  ( ( ( G  e. USGraph  /\  M  e.  V )  /\  ( C  e.  E  /\  M  e.  C )
)  ->  G  e. USGraph  )
3 edgnbusgreu.e . . . . . . . 8  |-  E  =  (Edg `  G )
43eleq2i 2541 . . . . . . 7  |-  ( C  e.  E  <->  C  e.  (Edg `  G ) )
54biimpi 199 . . . . . 6  |-  ( C  e.  E  ->  C  e.  (Edg `  G )
)
65ad2antrl 742 . . . . 5  |-  ( ( ( G  e. USGraph  /\  M  e.  V )  /\  ( C  e.  E  /\  M  e.  C )
)  ->  C  e.  (Edg `  G ) )
7 simprr 774 . . . . 5  |-  ( ( ( G  e. USGraph  /\  M  e.  V )  /\  ( C  e.  E  /\  M  e.  C )
)  ->  M  e.  C )
8 usgredg2vtxeu 39462 . . . . 5  |-  ( ( G  e. USGraph  /\  C  e.  (Edg `  G )  /\  M  e.  C
)  ->  E! n  e.  (Vtx `  G ) C  =  { M ,  n } )
92, 6, 7, 8syl3anc 1292 . . . 4  |-  ( ( ( G  e. USGraph  /\  M  e.  V )  /\  ( C  e.  E  /\  M  e.  C )
)  ->  E! n  e.  (Vtx `  G ) C  =  { M ,  n } )
10 df-reu 2763 . . . . 5  |-  ( E! n  e.  (Vtx `  G ) C  =  { M ,  n } 
<->  E! n ( n  e.  (Vtx `  G
)  /\  C  =  { M ,  n }
) )
11 prcom 4041 . . . . . . . . . . . . . . . 16  |-  { M ,  n }  =  {
n ,  M }
1211eqeq2i 2483 . . . . . . . . . . . . . . 15  |-  ( C  =  { M ,  n }  <->  C  =  {
n ,  M }
)
1312biimpi 199 . . . . . . . . . . . . . 14  |-  ( C  =  { M ,  n }  ->  C  =  { n ,  M } )
1413eleq1d 2533 . . . . . . . . . . . . 13  |-  ( C  =  { M ,  n }  ->  ( C  e.  E  <->  { n ,  M }  e.  E
) )
1514biimpcd 232 . . . . . . . . . . . 12  |-  ( C  e.  E  ->  ( C  =  { M ,  n }  ->  { n ,  M }  e.  E
) )
1615ad2antrl 742 . . . . . . . . . . 11  |-  ( ( ( G  e. USGraph  /\  M  e.  V )  /\  ( C  e.  E  /\  M  e.  C )
)  ->  ( C  =  { M ,  n }  ->  { n ,  M }  e.  E
) )
1716adantld 474 . . . . . . . . . 10  |-  ( ( ( G  e. USGraph  /\  M  e.  V )  /\  ( C  e.  E  /\  M  e.  C )
)  ->  ( (
n  e.  (Vtx `  G )  /\  C  =  { M ,  n } )  ->  { n ,  M }  e.  E
) )
1817imp 436 . . . . . . . . 9  |-  ( ( ( ( G  e. USGraph  /\  M  e.  V
)  /\  ( C  e.  E  /\  M  e.  C ) )  /\  ( n  e.  (Vtx `  G )  /\  C  =  { M ,  n } ) )  ->  { n ,  M }  e.  E )
19 simprr 774 . . . . . . . . 9  |-  ( ( ( ( G  e. USGraph  /\  M  e.  V
)  /\  ( C  e.  E  /\  M  e.  C ) )  /\  ( n  e.  (Vtx `  G )  /\  C  =  { M ,  n } ) )  ->  C  =  { M ,  n } )
2018, 19jca 541 . . . . . . . 8  |-  ( ( ( ( G  e. USGraph  /\  M  e.  V
)  /\  ( C  e.  E  /\  M  e.  C ) )  /\  ( n  e.  (Vtx `  G )  /\  C  =  { M ,  n } ) )  -> 
( { n ,  M }  e.  E  /\  C  =  { M ,  n }
) )
21 simpl 464 . . . . . . . . . 10  |-  ( ( { n ,  M }  e.  E  /\  C  =  { M ,  n } )  ->  { n ,  M }  e.  E )
22 eqid 2471 . . . . . . . . . . . 12  |-  (Vtx `  G )  =  (Vtx
`  G )
233, 22usgrpredgav 39442 . . . . . . . . . . 11  |-  ( ( G  e. USGraph  /\  { n ,  M }  e.  E
)  ->  ( n  e.  (Vtx `  G )  /\  M  e.  (Vtx `  G ) ) )
2423simpld 466 . . . . . . . . . 10  |-  ( ( G  e. USGraph  /\  { n ,  M }  e.  E
)  ->  n  e.  (Vtx `  G ) )
252, 21, 24syl2an 485 . . . . . . . . 9  |-  ( ( ( ( G  e. USGraph  /\  M  e.  V
)  /\  ( C  e.  E  /\  M  e.  C ) )  /\  ( { n ,  M }  e.  E  /\  C  =  { M ,  n } ) )  ->  n  e.  (Vtx
`  G ) )
26 simprr 774 . . . . . . . . 9  |-  ( ( ( ( G  e. USGraph  /\  M  e.  V
)  /\  ( C  e.  E  /\  M  e.  C ) )  /\  ( { n ,  M }  e.  E  /\  C  =  { M ,  n } ) )  ->  C  =  { M ,  n }
)
2725, 26jca 541 . . . . . . . 8  |-  ( ( ( ( G  e. USGraph  /\  M  e.  V
)  /\  ( C  e.  E  /\  M  e.  C ) )  /\  ( { n ,  M }  e.  E  /\  C  =  { M ,  n } ) )  ->  ( n  e.  (Vtx `  G )  /\  C  =  { M ,  n }
) )
2820, 27impbida 850 . . . . . . 7  |-  ( ( ( G  e. USGraph  /\  M  e.  V )  /\  ( C  e.  E  /\  M  e.  C )
)  ->  ( (
n  e.  (Vtx `  G )  /\  C  =  { M ,  n } )  <->  ( {
n ,  M }  e.  E  /\  C  =  { M ,  n } ) ) )
2928eubidv 2339 . . . . . 6  |-  ( ( ( G  e. USGraph  /\  M  e.  V )  /\  ( C  e.  E  /\  M  e.  C )
)  ->  ( E! n ( n  e.  (Vtx `  G )  /\  C  =  { M ,  n }
)  <->  E! n ( { n ,  M }  e.  E  /\  C  =  { M ,  n } ) ) )
3029biimpd 212 . . . . 5  |-  ( ( ( G  e. USGraph  /\  M  e.  V )  /\  ( C  e.  E  /\  M  e.  C )
)  ->  ( E! n ( n  e.  (Vtx `  G )  /\  C  =  { M ,  n }
)  ->  E! n
( { n ,  M }  e.  E  /\  C  =  { M ,  n }
) ) )
3110, 30syl5bi 225 . . . 4  |-  ( ( ( G  e. USGraph  /\  M  e.  V )  /\  ( C  e.  E  /\  M  e.  C )
)  ->  ( E! n  e.  (Vtx `  G
) C  =  { M ,  n }  ->  E! n ( { n ,  M }  e.  E  /\  C  =  { M ,  n } ) ) )
329, 31mpd 15 . . 3  |-  ( ( ( G  e. USGraph  /\  M  e.  V )  /\  ( C  e.  E  /\  M  e.  C )
)  ->  E! n
( { n ,  M }  e.  E  /\  C  =  { M ,  n }
) )
33 edgnbusgreu.n . . . . . . . 8  |-  N  =  ( G NeighbVtx  M )
3433eleq2i 2541 . . . . . . 7  |-  ( n  e.  N  <->  n  e.  ( G NeighbVtx  M ) )
353nbusgreledg 39585 . . . . . . 7  |-  ( G  e. USGraph  ->  ( n  e.  ( G NeighbVtx  M )  <->  { n ,  M }  e.  E ) )
3634, 35syl5bb 265 . . . . . 6  |-  ( G  e. USGraph  ->  ( n  e.  N  <->  { n ,  M }  e.  E )
)
3736anbi1d 719 . . . . 5  |-  ( G  e. USGraph  ->  ( ( n  e.  N  /\  C  =  { M ,  n } )  <->  ( {
n ,  M }  e.  E  /\  C  =  { M ,  n } ) ) )
3837ad2antrr 740 . . . 4  |-  ( ( ( G  e. USGraph  /\  M  e.  V )  /\  ( C  e.  E  /\  M  e.  C )
)  ->  ( (
n  e.  N  /\  C  =  { M ,  n } )  <->  ( {
n ,  M }  e.  E  /\  C  =  { M ,  n } ) ) )
3938eubidv 2339 . . 3  |-  ( ( ( G  e. USGraph  /\  M  e.  V )  /\  ( C  e.  E  /\  M  e.  C )
)  ->  ( E! n ( n  e.  N  /\  C  =  { M ,  n } )  <->  E! n
( { n ,  M }  e.  E  /\  C  =  { M ,  n }
) ) )
4032, 39mpbird 240 . 2  |-  ( ( ( G  e. USGraph  /\  M  e.  V )  /\  ( C  e.  E  /\  M  e.  C )
)  ->  E! n
( n  e.  N  /\  C  =  { M ,  n }
) )
41 df-reu 2763 . 2  |-  ( E! n  e.  N  C  =  { M ,  n } 
<->  E! n ( n  e.  N  /\  C  =  { M ,  n } ) )
4240, 41sylibr 217 1  |-  ( ( ( G  e. USGraph  /\  M  e.  V )  /\  ( C  e.  E  /\  M  e.  C )
)  ->  E! n  e.  N  C  =  { M ,  n }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   E!weu 2319   E!wreu 2758   {cpr 3961   ` cfv 5589  (class class class)co 6308  Vtxcvtx 39251  Edgcedga 39371   USGraph cusgr 39397   NeighbVtx cnbgr 39561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-upgr 39328  df-umgr 39329  df-edga 39372  df-uspgr 39398  df-usgr 39399  df-nbgr 39565
This theorem is referenced by:  nbusgrf1o0  39607
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