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Theorem usgra2edg 25109
Description: If a vertex is adjacent to two different vertices in a simple graph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.)
Assertion
Ref Expression
usgra2edg  |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  /\  ( { N ,  b }  e.  ran  E  /\  { c ,  N }  e.  ran  E ) )  ->  E. x  e.  dom  E E. y  e.  dom  E ( x  =/=  y  /\  N  e.  ( E `  x
)  /\  N  e.  ( E `  y ) ) )
Distinct variable groups:    E, b,
c, x, y    N, b, c, x, y    x, V, y
Allowed substitution hints:    V( b, c)

Proof of Theorem usgra2edg
StepHypRef Expression
1 usgrafun 25076 . . . . . 6  |-  ( V USGrph  E  ->  Fun  E )
2 funfn 5611 . . . . . 6  |-  ( Fun 
E  <->  E  Fn  dom  E )
31, 2sylib 200 . . . . 5  |-  ( V USGrph  E  ->  E  Fn  dom  E )
433ad2ant1 1029 . . . 4  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  E  Fn  dom  E )
5 fvelrnb 5912 . . . . 5  |-  ( E  Fn  dom  E  -> 
( { N , 
b }  e.  ran  E  <->  E. x  e.  dom  E ( E `  x
)  =  { N ,  b } ) )
6 fvelrnb 5912 . . . . 5  |-  ( E  Fn  dom  E  -> 
( { c ,  N }  e.  ran  E  <->  E. y  e.  dom  E ( E `  y
)  =  { c ,  N } ) )
75, 6anbi12d 717 . . . 4  |-  ( E  Fn  dom  E  -> 
( ( { N ,  b }  e.  ran  E  /\  { c ,  N }  e.  ran  E )  <->  ( E. x  e.  dom  E ( E `  x )  =  { N , 
b }  /\  E. y  e.  dom  E ( E `  y )  =  { c ,  N } ) ) )
84, 7syl 17 . . 3  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  (
( { N , 
b }  e.  ran  E  /\  { c ,  N }  e.  ran  E )  <->  ( E. x  e.  dom  E ( E `
 x )  =  { N ,  b }  /\  E. y  e.  dom  E ( E `
 y )  =  { c ,  N } ) ) )
9 reeanv 2958 . . . 4  |-  ( E. x  e.  dom  E E. y  e.  dom  E ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } )  <-> 
( E. x  e. 
dom  E ( E `
 x )  =  { N ,  b }  /\  E. y  e.  dom  E ( E `
 y )  =  { c ,  N } ) )
10 fveq2 5865 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  ( E `  x )  =  ( E `  y ) )
1110eqeq1d 2453 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  (
( E `  x
)  =  { N ,  b }  <->  ( E `  y )  =  { N ,  b }
) )
1211anbi1d 711 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } )  <-> 
( ( E `  y )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) ) )
13 eqtr2 2471 . . . . . . . . . . . . . 14  |-  ( ( ( E `  y
)  =  { N ,  b }  /\  ( E `  y )  =  { c ,  N } )  ->  { N ,  b }  =  { c ,  N } )
14 prcom 4050 . . . . . . . . . . . . . . . 16  |-  { c ,  N }  =  { N ,  c }
1514eqeq2i 2463 . . . . . . . . . . . . . . 15  |-  ( { N ,  b }  =  { c ,  N }  <->  { N ,  b }  =  { N ,  c } )
16 vex 3048 . . . . . . . . . . . . . . . . 17  |-  b  e. 
_V
17 vex 3048 . . . . . . . . . . . . . . . . 17  |-  c  e. 
_V
1816, 17preqr2 4150 . . . . . . . . . . . . . . . 16  |-  ( { N ,  b }  =  { N , 
c }  ->  b  =  c )
19 eqneqall 2634 . . . . . . . . . . . . . . . 16  |-  ( b  =  c  ->  (
b  =/=  c  ->  x  =/=  y ) )
2018, 19syl 17 . . . . . . . . . . . . . . 15  |-  ( { N ,  b }  =  { N , 
c }  ->  (
b  =/=  c  ->  x  =/=  y ) )
2115, 20sylbi 199 . . . . . . . . . . . . . 14  |-  ( { N ,  b }  =  { c ,  N }  ->  (
b  =/=  c  ->  x  =/=  y ) )
2213, 21syl 17 . . . . . . . . . . . . 13  |-  ( ( ( E `  y
)  =  { N ,  b }  /\  ( E `  y )  =  { c ,  N } )  -> 
( b  =/=  c  ->  x  =/=  y ) )
2312, 22syl6bi 232 . . . . . . . . . . . 12  |-  ( x  =  y  ->  (
( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } )  ->  ( b  =/=  c  ->  x  =/=  y ) ) )
2423com23 81 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
b  =/=  c  -> 
( ( ( E `
 x )  =  { N ,  b }  /\  ( E `
 y )  =  { c ,  N } )  ->  x  =/=  y ) ) )
2524impd 433 . . . . . . . . . 10  |-  ( x  =  y  ->  (
( b  =/=  c  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  x  =/=  y ) )
26 ax-1 6 . . . . . . . . . 10  |-  ( x  =/=  y  ->  (
( b  =/=  c  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  x  =/=  y ) )
2725, 26pm2.61ine 2707 . . . . . . . . 9  |-  ( ( b  =/=  c  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  x  =/=  y )
28273ad2antl3 1172 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  x  =/=  y )
29 prid1g 4078 . . . . . . . . . . . 12  |-  ( N  e.  V  ->  N  e.  { N ,  b } )
30293ad2ant2 1030 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  N  e.  { N ,  b } )
31 eleq2 2518 . . . . . . . . . . 11  |-  ( ( E `  x )  =  { N , 
b }  ->  ( N  e.  ( E `  x )  <->  N  e.  { N ,  b } ) )
3230, 31syl5ibr 225 . . . . . . . . . 10  |-  ( ( E `  x )  =  { N , 
b }  ->  (
( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  N  e.  ( E `  x ) ) )
3332adantr 467 . . . . . . . . 9  |-  ( ( ( E `  x
)  =  { N ,  b }  /\  ( E `  y )  =  { c ,  N } )  -> 
( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c
)  ->  N  e.  ( E `  x ) ) )
3433impcom 432 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  N  e.  ( E `  x ) )
35 prid2g 4079 . . . . . . . . . . . 12  |-  ( N  e.  V  ->  N  e.  { c ,  N } )
36353ad2ant2 1030 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  N  e.  { c ,  N } )
37 eleq2 2518 . . . . . . . . . . 11  |-  ( ( E `  y )  =  { c ,  N }  ->  ( N  e.  ( E `  y )  <->  N  e.  { c ,  N }
) )
3836, 37syl5ibr 225 . . . . . . . . . 10  |-  ( ( E `  y )  =  { c ,  N }  ->  (
( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  N  e.  ( E `  y ) ) )
3938adantl 468 . . . . . . . . 9  |-  ( ( ( E `  x
)  =  { N ,  b }  /\  ( E `  y )  =  { c ,  N } )  -> 
( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c
)  ->  N  e.  ( E `  y ) ) )
4039impcom 432 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  N  e.  ( E `  y ) )
4128, 34, 403jca 1188 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  ( x  =/=  y  /\  N  e.  ( E `  x
)  /\  N  e.  ( E `  y ) ) )
4241ex 436 . . . . . 6  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  (
( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } )  ->  ( x  =/=  y  /\  N  e.  ( E `  x
)  /\  N  e.  ( E `  y ) ) ) )
4342reximdv 2861 . . . . 5  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  ( E. y  e.  dom  E ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } )  ->  E. y  e.  dom  E ( x  =/=  y  /\  N  e.  ( E `  x )  /\  N  e.  ( E `  y )
) ) )
4443reximdv 2861 . . . 4  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  ( E. x  e.  dom  E E. y  e.  dom  E ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } )  ->  E. x  e.  dom  E E. y  e.  dom  E ( x  =/=  y  /\  N  e.  ( E `  x )  /\  N  e.  ( E `  y )
) ) )
459, 44syl5bir 222 . . 3  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  (
( E. x  e. 
dom  E ( E `
 x )  =  { N ,  b }  /\  E. y  e.  dom  E ( E `
 y )  =  { c ,  N } )  ->  E. x  e.  dom  E E. y  e.  dom  E ( x  =/=  y  /\  N  e.  ( E `  x
)  /\  N  e.  ( E `  y ) ) ) )
468, 45sylbid 219 . 2  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  (
( { N , 
b }  e.  ran  E  /\  { c ,  N }  e.  ran  E )  ->  E. x  e.  dom  E E. y  e.  dom  E ( x  =/=  y  /\  N  e.  ( E `  x
)  /\  N  e.  ( E `  y ) ) ) )
4746imp 431 1  |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  /\  ( { N ,  b }  e.  ran  E  /\  { c ,  N }  e.  ran  E ) )  ->  E. x  e.  dom  E E. y  e.  dom  E ( x  =/=  y  /\  N  e.  ( E `  x
)  /\  N  e.  ( E `  y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   E.wrex 2738   {cpr 3970   class class class wbr 4402   dom cdm 4834   ran crn 4835   Fun wfun 5576    Fn wfn 5577   ` cfv 5582   USGrph cusg 25057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-hash 12516  df-usgra 25060
This theorem is referenced by:  usgra2edg1  25110
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