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Theorem umgr2v2enb1 39612
Description: In a multigraph with two edges connecting the same two vertices, each of the vertices has one neighbor. (Contributed by AV, 18-Dec-2020.)
Hypothesis
Ref Expression
umgr2v2evtx.g  |-  G  = 
<. V ,  { <. 0 ,  { A ,  B } >. ,  <. 1 ,  { A ,  B } >. } >.
Assertion
Ref Expression
umgr2v2enb1  |-  ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  ( G NeighbVtx  A )  =  { B } )

Proof of Theorem umgr2v2enb1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 umgr2v2evtx.g . . . 4  |-  G  = 
<. V ,  { <. 0 ,  { A ,  B } >. ,  <. 1 ,  { A ,  B } >. } >.
21umgr2v2e 39611 . . 3  |-  ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  G  e. UMGraph  )
31umgr2v2evtxel 39608 . . . . 5  |-  ( ( V  e.  W  /\  A  e.  V )  ->  A  e.  (Vtx `  G ) )
433adant3 1034 . . . 4  |-  ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V )  ->  A  e.  (Vtx `  G ) )
54adantr 471 . . 3  |-  ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  A  e.  (Vtx `  G ) )
6 eqid 2461 . . . 4  |-  (Vtx `  G )  =  (Vtx
`  G )
7 eqid 2461 . . . 4  |-  (Edg `  G )  =  (Edg
`  G )
86, 7nbumgrvtx 39463 . . 3  |-  ( ( G  e. UMGraph  /\  A  e.  (Vtx `  G )
)  ->  ( G NeighbVtx  A )  =  { x  e.  (Vtx `  G )  |  { A ,  x }  e.  (Edg `  G
) } )
92, 5, 8syl2anc 671 . 2  |-  ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  ( G NeighbVtx  A )  =  { x  e.  (Vtx `  G )  |  { A ,  x }  e.  (Edg `  G
) } )
101umgr2v2eedg 39610 . . . . . . . . . . 11  |-  ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V )  ->  (Edg `  G )  =  { { A ,  B } } )
1110eleq2d 2524 . . . . . . . . . 10  |-  ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V )  ->  ( { A ,  x }  e.  (Edg `  G )  <->  { A ,  x }  e.  { { A ,  B } } ) )
1211adantr 471 . . . . . . . . 9  |-  ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  ( { A ,  x }  e.  (Edg `  G )  <->  { A ,  x }  e.  { { A ,  B } } ) )
1312adantr 471 . . . . . . . 8  |-  ( ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V )  /\  A  =/=  B )  /\  x  e.  (Vtx `  G )
)  ->  ( { A ,  x }  e.  (Edg `  G )  <->  { A ,  x }  e.  { { A ,  B } } ) )
14 prex 4655 . . . . . . . . 9  |-  { A ,  x }  e.  _V
1514elsnc 4003 . . . . . . . 8  |-  ( { A ,  x }  e.  { { A ,  B } }  <->  { A ,  x }  =  { A ,  B }
)
1613, 15syl6bb 269 . . . . . . 7  |-  ( ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V )  /\  A  =/=  B )  /\  x  e.  (Vtx `  G )
)  ->  ( { A ,  x }  e.  (Edg `  G )  <->  { A ,  x }  =  { A ,  B } ) )
17 simpr 467 . . . . . . . 8  |-  ( ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V )  /\  A  =/=  B )  /\  x  e.  (Vtx `  G )
)  ->  x  e.  (Vtx `  G ) )
18 simpll3 1055 . . . . . . . 8  |-  ( ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V )  /\  A  =/=  B )  /\  x  e.  (Vtx `  G )
)  ->  B  e.  V )
1917, 18preq2b 4159 . . . . . . 7  |-  ( ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V )  /\  A  =/=  B )  /\  x  e.  (Vtx `  G )
)  ->  ( { A ,  x }  =  { A ,  B } 
<->  x  =  B ) )
2016, 19bitrd 261 . . . . . 6  |-  ( ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V )  /\  A  =/=  B )  /\  x  e.  (Vtx `  G )
)  ->  ( { A ,  x }  e.  (Edg `  G )  <->  x  =  B ) )
2120pm5.32da 651 . . . . 5  |-  ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  ( (
x  e.  (Vtx `  G )  /\  { A ,  x }  e.  (Edg `  G )
)  <->  ( x  e.  (Vtx `  G )  /\  x  =  B
) ) )
221umgr2v2evtx 39607 . . . . . . . . 9  |-  ( V  e.  W  ->  (Vtx `  G )  =  V )
23223ad2ant1 1035 . . . . . . . 8  |-  ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V )  ->  (Vtx `  G )  =  V )
24 eleq12 2529 . . . . . . . . . . 11  |-  ( ( x  =  B  /\  (Vtx `  G )  =  V )  ->  (
x  e.  (Vtx `  G )  <->  B  e.  V ) )
2524exbiri 632 . . . . . . . . . 10  |-  ( x  =  B  ->  (
(Vtx `  G )  =  V  ->  ( B  e.  V  ->  x  e.  (Vtx `  G )
) ) )
2625com13 83 . . . . . . . . 9  |-  ( B  e.  V  ->  (
(Vtx `  G )  =  V  ->  ( x  =  B  ->  x  e.  (Vtx `  G )
) ) )
27263ad2ant3 1037 . . . . . . . 8  |-  ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V )  ->  ( (Vtx `  G
)  =  V  -> 
( x  =  B  ->  x  e.  (Vtx
`  G ) ) ) )
2823, 27mpd 15 . . . . . . 7  |-  ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V )  ->  ( x  =  B  ->  x  e.  (Vtx
`  G ) ) )
2928adantr 471 . . . . . 6  |-  ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  ( x  =  B  ->  x  e.  (Vtx `  G )
) )
3029pm4.71rd 645 . . . . 5  |-  ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  ( x  =  B  <->  ( x  e.  (Vtx `  G )  /\  x  =  B
) ) )
3121, 30bitr4d 264 . . . 4  |-  ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  ( (
x  e.  (Vtx `  G )  /\  { A ,  x }  e.  (Edg `  G )
)  <->  x  =  B
) )
3231alrimiv 1783 . . 3  |-  ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  A. x
( ( x  e.  (Vtx `  G )  /\  { A ,  x }  e.  (Edg `  G
) )  <->  x  =  B ) )
33 rabeqsn 4012 . . 3  |-  ( { x  e.  (Vtx `  G )  |  { A ,  x }  e.  (Edg `  G ) }  =  { B } 
<-> 
A. x ( ( x  e.  (Vtx `  G )  /\  { A ,  x }  e.  (Edg `  G )
)  <->  x  =  B
) )
3432, 33sylibr 217 . 2  |-  ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  { x  e.  (Vtx `  G )  |  { A ,  x }  e.  (Edg `  G
) }  =  { B } )
359, 34eqtrd 2495 1  |-  ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  ( G NeighbVtx  A )  =  { B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991   A.wal 1452    = wceq 1454    e. wcel 1897    =/= wne 2632   {crab 2752   {csn 3979   {cpr 3981   <.cop 3985   ` cfv 5600  (class class class)co 6314   0cc0 9564   1c1 9565  Vtxcvtx 39146   UMGraph cumgr 39219  Edgcedga 39257   NeighbVtx cnbgr 39446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-2o 7208  df-oadd 7211  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-card 8398  df-cda 8623  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-n0 10898  df-z 10966  df-uz 11188  df-fz 11813  df-hash 12547  df-vtx 39148  df-iedg 39149  df-upgr 39220  df-umgr 39221  df-edga 39258  df-nbgr 39450
This theorem is referenced by: (None)
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