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Theorem usgra2adedglem1 30476
Description: In an undirected simple graph, two adjacent edges are an unordered pair of unordered pairs. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
Hypotheses
Ref Expression
usgra2adedgspth.f  |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. }
usgra2adedgspth.p  |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }
Assertion
Ref Expression
usgra2adedglem1  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  { { A ,  B } ,  { B ,  C } }  =  ( E " ran  F
) ) )

Proof of Theorem usgra2adedglem1
StepHypRef Expression
1 usgraf1o 23453 . 2  |-  ( V USGrph  E  ->  E : dom  E -1-1-onto-> ran 
E )
2 usgra2adedgspth.f . . . . . . . 8  |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. }
32rneqi 5177 . . . . . . 7  |-  ran  F  =  ran  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. }
4 c0ex 9494 . . . . . . . 8  |-  0  e.  _V
5 1ex 9495 . . . . . . . 8  |-  1  e.  _V
6 rnpropg 5430 . . . . . . . 8  |-  ( ( 0  e.  _V  /\  1  e.  _V )  ->  ran  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. }  =  { ( `' E `  { A ,  B } ) ,  ( `' E `  { B ,  C }
) } )
74, 5, 6mp2an 672 . . . . . . 7  |-  ran  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. }  =  { ( `' E `  { A ,  B } ) ,  ( `' E `  { B ,  C }
) }
83, 7eqtri 2483 . . . . . 6  |-  ran  F  =  { ( `' E `  { A ,  B } ) ,  ( `' E `  { B ,  C } ) }
98a1i 11 . . . . 5  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ran  F  =  { ( `' E `  { A ,  B } ) ,  ( `' E `  { B ,  C }
) } )
109imaeq2d 5280 . . . 4  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( E " ran  F )  =  ( E " { ( `' E `  { A ,  B } ) ,  ( `' E `  { B ,  C } ) } ) )
11 f1ofn 5753 . . . . . 6  |-  ( E : dom  E -1-1-onto-> ran  E  ->  E  Fn  dom  E
)
1211adantr 465 . . . . 5  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  E  Fn  dom  E )
13 f1ocnvdm 6101 . . . . . 6  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { A ,  B }  e.  ran  E )  ->  ( `' E `  { A ,  B } )  e. 
dom  E )
1413adantrr 716 . . . . 5  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( `' E `  { A ,  B } )  e. 
dom  E )
15 f1ocnvdm 6101 . . . . . 6  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { B ,  C }  e.  ran  E )  ->  ( `' E `  { B ,  C } )  e. 
dom  E )
1615adantrl 715 . . . . 5  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( `' E `  { B ,  C } )  e. 
dom  E )
17 fnimapr 5867 . . . . 5  |-  ( ( E  Fn  dom  E  /\  ( `' E `  { A ,  B }
)  e.  dom  E  /\  ( `' E `  { B ,  C }
)  e.  dom  E
)  ->  ( E " { ( `' E `  { A ,  B } ) ,  ( `' E `  { B ,  C } ) } )  =  { ( E `  ( `' E `  { A ,  B } ) ) ,  ( E `  ( `' E `  { B ,  C } ) ) } )
1812, 14, 16, 17syl3anc 1219 . . . 4  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( E " { ( `' E `  { A ,  B } ) ,  ( `' E `  { B ,  C }
) } )  =  { ( E `  ( `' E `  { A ,  B } ) ) ,  ( E `  ( `' E `  { B ,  C } ) ) } )
19 f1ocnvfv2 6096 . . . . . 6  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { A ,  B }  e.  ran  E )  ->  ( E `  ( `' E `  { A ,  B }
) )  =  { A ,  B }
)
2019adantrr 716 . . . . 5  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B } )
21 f1ocnvfv2 6096 . . . . . 6  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { B ,  C }  e.  ran  E )  ->  ( E `  ( `' E `  { B ,  C }
) )  =  { B ,  C }
)
2221adantrl 715 . . . . 5  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } )
2320, 22preq12d 4073 . . . 4  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  { ( E `  ( `' E `  { A ,  B } ) ) ,  ( E `  ( `' E `  { B ,  C } ) ) }  =  { { A ,  B } ,  { B ,  C } } )
2410, 18, 233eqtrrd 2500 . . 3  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  { { A ,  B } ,  { B ,  C } }  =  ( E " ran  F ) )
2524ex 434 . 2  |-  ( E : dom  E -1-1-onto-> ran  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  { { A ,  B } ,  { B ,  C } }  =  ( E " ran  F ) ) )
261, 25syl 16 1  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  { { A ,  B } ,  { B ,  C } }  =  ( E " ran  F
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   {cpr 3990   {ctp 3992   <.cop 3994   class class class wbr 4403   `'ccnv 4950   dom cdm 4951   ran crn 4952   "cima 4954    Fn wfn 5524   -1-1-onto->wf1o 5528   ` cfv 5529   0cc0 9396   1c1 9397   2c2 10485   USGrph cusg 23436
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 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
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-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-hash 12224  df-usgra 23438
This theorem is referenced by: (None)
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