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Theorem usgra2adedgspthlem2 30453
Description: Lemma for usgra2adedgspth 30454. (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 23434 . . . 4  |-  ( V USGrph  E  ->  E : dom  E -1-1-onto-> ran 
E )
2 f1of1 5749 . . . . 5  |-  ( E : dom  E -1-1-onto-> ran  E  ->  E : dom  E -1-1-> ran 
E )
3 f1ocnvfvrneq 6100 . . . . . . . . . 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 23449 . . . . . . . . . . . . . . . . . . 19  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  ( B  e.  V  /\  C  e.  V ) )
6 usgraedgrnv 23449 . . . . . . . . . . . . . . . . . . . . . 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 4160 . . . . . . . . . . 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 2480 . . . . . . . . . . 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 2676 . . . . . . 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 30452 . . 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 969 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   {cpr 3988   class class class wbr 4401   `'ccnv 4948   dom cdm 4949   ran crn 4950   -1-1->wf1 5524   -1-1-onto->wf1o 5526   ` cfv 5527   USGrph cusg 23417
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 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
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 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-hash 12222  df-usgra 23419
This theorem is referenced by:  usgra2adedgspth  30454
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