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Theorem usgrarnedg 24088
Description: For each edge in a simple graph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.)
Assertion
Ref Expression
usgrarnedg  |-  ( ( V USGrph  E  /\  Y  e. 
ran  E )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  {
a ,  b } ) )
Distinct variable groups:    E, a,
b    Y, a, b    V, a, b

Proof of Theorem usgrarnedg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrafun 24053 . . . 4  |-  ( V USGrph  E  ->  Fun  E )
2 funfn 5617 . . . . 5  |-  ( Fun 
E  <->  E  Fn  dom  E )
32biimpi 194 . . . 4  |-  ( Fun 
E  ->  E  Fn  dom  E )
4 fvelrnb 5915 . . . 4  |-  ( E  Fn  dom  E  -> 
( Y  e.  ran  E  <->  E. x  e.  dom  E ( E `  x
)  =  Y ) )
51, 3, 43syl 20 . . 3  |-  ( V USGrph  E  ->  ( Y  e. 
ran  E  <->  E. x  e.  dom  E ( E `  x
)  =  Y ) )
6 usgraf0 24052 . . . . 5  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { y  e.  ~P V  |  ( # `  y
)  =  2 } )
7 f1f 5781 . . . . 5  |-  ( E : dom  E -1-1-> {
y  e.  ~P V  |  ( # `  y
)  =  2 }  ->  E : dom  E --> { y  e.  ~P V  |  ( # `  y
)  =  2 } )
8 ffvelrn 6019 . . . . . . 7  |-  ( ( E : dom  E --> { y  e.  ~P V  |  ( # `  y
)  =  2 }  /\  x  e.  dom  E )  ->  ( E `  x )  e.  {
y  e.  ~P V  |  ( # `  y
)  =  2 } )
9 eleq1 2539 . . . . . . . 8  |-  ( ( E `  x )  =  Y  ->  (
( E `  x
)  e.  { y  e.  ~P V  | 
( # `  y )  =  2 }  <->  Y  e.  { y  e.  ~P V  |  ( # `  y
)  =  2 } ) )
10 fveq2 5866 . . . . . . . . . . 11  |-  ( y  =  Y  ->  ( # `
 y )  =  ( # `  Y
) )
1110eqeq1d 2469 . . . . . . . . . 10  |-  ( y  =  Y  ->  (
( # `  y )  =  2  <->  ( # `  Y
)  =  2 ) )
1211elrab 3261 . . . . . . . . 9  |-  ( Y  e.  { y  e. 
~P V  |  (
# `  y )  =  2 }  <->  ( Y  e.  ~P V  /\  ( # `
 Y )  =  2 ) )
13 hash2prde 12482 . . . . . . . . . 10  |-  ( ( Y  e.  ~P V  /\  ( # `  Y
)  =  2 )  ->  E. a E. b
( a  =/=  b  /\  Y  =  {
a ,  b } ) )
14 eleq1 2539 . . . . . . . . . . . . . . . . 17  |-  ( Y  =  { a ,  b }  ->  ( Y  e.  ~P V  <->  { a ,  b }  e.  ~P V ) )
15 prex 4689 . . . . . . . . . . . . . . . . . . 19  |-  { a ,  b }  e.  _V
1615elpw 4016 . . . . . . . . . . . . . . . . . 18  |-  ( { a ,  b }  e.  ~P V  <->  { a ,  b }  C_  V )
17 vex 3116 . . . . . . . . . . . . . . . . . . . 20  |-  a  e. 
_V
18 vex 3116 . . . . . . . . . . . . . . . . . . . 20  |-  b  e. 
_V
1917, 18prss 4181 . . . . . . . . . . . . . . . . . . 19  |-  ( ( a  e.  V  /\  b  e.  V )  <->  { a ,  b } 
C_  V )
2019biimpri 206 . . . . . . . . . . . . . . . . . 18  |-  ( { a ,  b } 
C_  V  ->  (
a  e.  V  /\  b  e.  V )
)
2116, 20sylbi 195 . . . . . . . . . . . . . . . . 17  |-  ( { a ,  b }  e.  ~P V  -> 
( a  e.  V  /\  b  e.  V
) )
2214, 21syl6bi 228 . . . . . . . . . . . . . . . 16  |-  ( Y  =  { a ,  b }  ->  ( Y  e.  ~P V  ->  ( a  e.  V  /\  b  e.  V
) ) )
2322adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( a  =/=  b  /\  Y  =  { a ,  b } )  ->  ( Y  e. 
~P V  ->  (
a  e.  V  /\  b  e.  V )
) )
2423imdistanri 691 . . . . . . . . . . . . . 14  |-  ( ( Y  e.  ~P V  /\  ( a  =/=  b  /\  Y  =  {
a ,  b } ) )  ->  (
( a  e.  V  /\  b  e.  V
)  /\  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) )
2524ex 434 . . . . . . . . . . . . 13  |-  ( Y  e.  ~P V  -> 
( ( a  =/=  b  /\  Y  =  { a ,  b } )  ->  (
( a  e.  V  /\  b  e.  V
)  /\  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) ) )
26252eximdv 1688 . . . . . . . . . . . 12  |-  ( Y  e.  ~P V  -> 
( E. a E. b ( a  =/=  b  /\  Y  =  { a ,  b } )  ->  E. a E. b ( ( a  e.  V  /\  b  e.  V )  /\  (
a  =/=  b  /\  Y  =  { a ,  b } ) ) ) )
27 r2ex 2985 . . . . . . . . . . . 12  |-  ( E. a  e.  V  E. b  e.  V  (
a  =/=  b  /\  Y  =  { a ,  b } )  <->  E. a E. b ( ( a  e.  V  /\  b  e.  V
)  /\  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) )
2826, 27syl6ibr 227 . . . . . . . . . . 11  |-  ( Y  e.  ~P V  -> 
( E. a E. b ( a  =/=  b  /\  Y  =  { a ,  b } )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) )
2928adantr 465 . . . . . . . . . 10  |-  ( ( Y  e.  ~P V  /\  ( # `  Y
)  =  2 )  ->  ( E. a E. b ( a  =/=  b  /\  Y  =  { a ,  b } )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) )
3013, 29mpd 15 . . . . . . . . 9  |-  ( ( Y  e.  ~P V  /\  ( # `  Y
)  =  2 )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  {
a ,  b } ) )
3112, 30sylbi 195 . . . . . . . 8  |-  ( Y  e.  { y  e. 
~P V  |  (
# `  y )  =  2 }  ->  E. a  e.  V  E. b  e.  V  (
a  =/=  b  /\  Y  =  { a ,  b } ) )
329, 31syl6bi 228 . . . . . . 7  |-  ( ( E `  x )  =  Y  ->  (
( E `  x
)  e.  { y  e.  ~P V  | 
( # `  y )  =  2 }  ->  E. a  e.  V  E. b  e.  V  (
a  =/=  b  /\  Y  =  { a ,  b } ) ) )
338, 32syl5com 30 . . . . . 6  |-  ( ( E : dom  E --> { y  e.  ~P V  |  ( # `  y
)  =  2 }  /\  x  e.  dom  E )  ->  ( ( E `  x )  =  Y  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) )
3433ex 434 . . . . 5  |-  ( E : dom  E --> { y  e.  ~P V  | 
( # `  y )  =  2 }  ->  ( x  e.  dom  E  ->  ( ( E `  x )  =  Y  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  {
a ,  b } ) ) ) )
356, 7, 343syl 20 . . . 4  |-  ( V USGrph  E  ->  ( x  e. 
dom  E  ->  ( ( E `  x )  =  Y  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) ) )
3635rexlimdv 2953 . . 3  |-  ( V USGrph  E  ->  ( E. x  e.  dom  E ( E `
 x )  =  Y  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) )
375, 36sylbid 215 . 2  |-  ( V USGrph  E  ->  ( Y  e. 
ran  E  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  { a ,  b } ) ) )
3837imp 429 1  |-  ( ( V USGrph  E  /\  Y  e. 
ran  E )  ->  E. a  e.  V  E. b  e.  V  ( a  =/=  b  /\  Y  =  {
a ,  b } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   E.wrex 2815   {crab 2818    C_ wss 3476   ~Pcpw 4010   {cpr 4029   class class class wbr 4447   dom cdm 4999   ran crn 5000   Fun wfun 5582    Fn wfn 5583   -->wf 5584   -1-1->wf1 5585   ` cfv 5588   2c2 10585   #chash 12373   USGrph cusg 24034
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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
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-rmo 2822  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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-hash 12374  df-usgra 24037
This theorem is referenced by:  edgprvtx  24089  usgraedg3  24090  usgrarnedg1  24093  usgrasscusgra  24187  sizeusglecusglem1  24188
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