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Theorem usgra2adedgspthlem2 24285
Description: Lemma 2 for usgra2adedgspth 24286. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
Assertion
Ref Expression
usgra2adedgspthlem2  |-  ( ( ( V USGrph  E  /\  A  =/=  C )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( ( `' E `  { A ,  B } )  =/=  ( `' E `  { B ,  C }
)  /\  ( E `  ( `' E `  { A ,  B }
) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } ) )

Proof of Theorem usgra2adedgspthlem2
StepHypRef Expression
1 usgraf1o 24031 . . . 4  |-  ( V USGrph  E  ->  E : dom  E -1-1-onto-> ran 
E )
2 f1of1 5813 . . . . 5  |-  ( E : dom  E -1-1-onto-> ran  E  ->  E : dom  E -1-1-> ran 
E )
3 f1ocnvfvrneq 6175 . . . . . . . . . 10  |-  ( ( E : dom  E -1-1-> ran 
E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( ( `' E `  { A ,  B } )  =  ( `' E `  { B ,  C }
)  ->  { A ,  B }  =  { B ,  C }
) )
43adantlr 714 . . . . . . . . 9  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  V USGrph  E )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( ( `' E `  { A ,  B } )  =  ( `' E `  { B ,  C }
)  ->  { A ,  B }  =  { B ,  C }
) )
5 usgraedgrnv 24050 . . . . . . . . . . . . . . . . . . 19  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  ( B  e.  V  /\  C  e.  V ) )
6 usgraedgrnv 24050 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  ( A  e.  V  /\  B  e.  V ) )
7 pm3.2 447 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( ( B  e.  V  /\  C  e.  V )  ->  (
( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) ) ) )
86, 7syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  ( ( B  e.  V  /\  C  e.  V )  ->  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) ) )
98ex 434 . . . . . . . . . . . . . . . . . . . 20  |-  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  ->  ( ( B  e.  V  /\  C  e.  V )  ->  (
( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) ) ) ) )
109com13 80 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  e.  V  /\  C  e.  V )  ->  ( { A ,  B }  e.  ran  E  ->  ( V USGrph  E  ->  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) ) ) )
115, 10syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  ( { A ,  B }  e.  ran  E  ->  ( V USGrph  E  ->  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V
) ) ) ) )
1211ex 434 . . . . . . . . . . . . . . . . 17  |-  ( V USGrph  E  ->  ( { B ,  C }  e.  ran  E  ->  ( { A ,  B }  e.  ran  E  ->  ( V USGrph  E  ->  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) ) ) ) )
1312com13 80 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B }  e.  ran  E  ->  ( { B ,  C }  e.  ran  E  ->  ( V USGrph  E  ->  ( V USGrph  E  ->  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) ) ) ) )
1413imp 429 . . . . . . . . . . . . . . 15  |-  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  -> 
( V USGrph  E  ->  ( V USGrph  E  ->  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V
) ) ) ) )
1514com13 80 . . . . . . . . . . . . . 14  |-  ( V USGrph  E  ->  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V
) ) ) ) )
1615pm2.43i 47 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  -> 
( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) ) )
1716adantl 466 . . . . . . . . . . . 12  |-  ( ( E : dom  E -1-1-> ran 
E  /\  V USGrph  E )  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  -> 
( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) ) )
1817imp 429 . . . . . . . . . . 11  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  V USGrph  E )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) )
19 preq12bg 4205 . . . . . . . . . . 11  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) )  -> 
( { A ,  B }  =  { B ,  C }  <->  ( ( A  =  B  /\  B  =  C )  \/  ( A  =  C  /\  B  =  B ) ) ) )
2018, 19syl 16 . . . . . . . . . 10  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  V USGrph  E )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( { A ,  B }  =  { B ,  C }  <->  ( ( A  =  B  /\  B  =  C )  \/  ( A  =  C  /\  B  =  B ) ) ) )
21 eqtr 2493 . . . . . . . . . . 11  |-  ( ( A  =  B  /\  B  =  C )  ->  A  =  C )
22 simpl 457 . . . . . . . . . . 11  |-  ( ( A  =  C  /\  B  =  B )  ->  A  =  C )
2321, 22jaoi 379 . . . . . . . . . 10  |-  ( ( ( A  =  B  /\  B  =  C )  \/  ( A  =  C  /\  B  =  B ) )  ->  A  =  C )
2420, 23syl6bi 228 . . . . . . . . 9  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  V USGrph  E )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( { A ,  B }  =  { B ,  C }  ->  A  =  C ) )
254, 24syld 44 . . . . . . . 8  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  V USGrph  E )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( ( `' E `  { A ,  B } )  =  ( `' E `  { B ,  C }
)  ->  A  =  C ) )
2625necon3d 2691 . . . . . . 7  |-  ( ( ( E : dom  E
-1-1-> ran  E  /\  V USGrph  E )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( A  =/= 
C  ->  ( `' E `  { A ,  B } )  =/=  ( `' E `  { B ,  C }
) ) )
2726exp31 604 . . . . . 6  |-  ( E : dom  E -1-1-> ran  E  ->  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  ( A  =/=  C  ->  ( `' E `  { A ,  B } )  =/=  ( `' E `  { B ,  C }
) ) ) ) )
2827com34 83 . . . . 5  |-  ( E : dom  E -1-1-> ran  E  ->  ( V USGrph  E  ->  ( A  =/=  C  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  ( `' E `  { A ,  B } )  =/=  ( `' E `  { B ,  C }
) ) ) ) )
292, 28syl 16 . . . 4  |-  ( E : dom  E -1-1-onto-> ran  E  ->  ( V USGrph  E  -> 
( A  =/=  C  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  ( `' E `  { A ,  B } )  =/=  ( `' E `  { B ,  C }
) ) ) ) )
301, 29mpcom 36 . . 3  |-  ( V USGrph  E  ->  ( A  =/= 
C  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  -> 
( `' E `  { A ,  B }
)  =/=  ( `' E `  { B ,  C } ) ) ) )
3130imp31 432 . 2  |-  ( ( ( V USGrph  E  /\  A  =/=  C )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( `' E `  { A ,  B } )  =/=  ( `' E `  { B ,  C } ) )
32 usgra2adedgspthlem1 24284 . . 3  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( ( E `
 ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } ) )
3332adantlr 714 . 2  |-  ( ( ( V USGrph  E  /\  A  =/=  C )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( ( E `
 ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } ) )
34 3anass 977 . 2  |-  ( ( ( `' E `  { A ,  B }
)  =/=  ( `' E `  { B ,  C } )  /\  ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } )  <->  ( ( `' E `  { A ,  B } )  =/=  ( `' E `  { B ,  C }
)  /\  ( ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } ) ) )
3531, 33, 34sylanbrc 664 1  |-  ( ( ( V USGrph  E  /\  A  =/=  C )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  ( ( `' E `  { A ,  B } )  =/=  ( `' E `  { B ,  C }
)  /\  ( E `  ( `' E `  { A ,  B }
) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   {cpr 4029   class class class wbr 4447   `'ccnv 4998   dom cdm 4999   ran crn 5000   -1-1->wf1 5583   -1-1-onto->wf1o 5585   ` cfv 5586   USGrph cusg 24003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-hash 12368  df-usgra 24006
This theorem is referenced by:  usgra2adedgspth  24286
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