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Theorem usgra2adedgspthlem2 24739
Description: Lemma 2 for usgra2adedgspth 24740. (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 24485 . . . 4  |-  ( V USGrph  E  ->  E : dom  E -1-1-onto-> ran 
E )
2 f1of1 5821 . . . . 5  |-  ( E : dom  E -1-1-onto-> ran  E  ->  E : dom  E -1-1-> ran 
E )
3 f1ocnvfvrneq 6190 . . . . . . . . . 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 24504 . . . . . . . . . . . . . . . . . . 19  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  ( B  e.  V  /\  C  e.  V ) )
6 usgraedgrnv 24504 . . . . . . . . . . . . . . . . . . . . . 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 4211 . . . . . . . . . . 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 2483 . . . . . . . . . . 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 2681 . . . . . . 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 24738 . . 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 1395    e. wcel 1819    =/= wne 2652   {cpr 4034   class class class wbr 4456   `'ccnv 5007   dom cdm 5008   ran crn 5009   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594   USGrph cusg 24457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-hash 12409  df-usgra 24460
This theorem is referenced by:  usgra2adedgspth  24740
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