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Theorem usgredg4 39458
Description: For a vertex incident to an edge there is another vertex incident to the edge. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 17-Oct-2020.)
Hypotheses
Ref Expression
usgredg3.v  |-  V  =  (Vtx `  G )
usgredg3.e  |-  E  =  (iEdg `  G )
Assertion
Ref Expression
usgredg4  |-  ( ( G  e. USGraph  /\  X  e. 
dom  E  /\  Y  e.  ( E `  X
) )  ->  E. y  e.  V  ( E `  X )  =  { Y ,  y }
)
Distinct variable groups:    y, E    y, G    y, V    y, X    y, Y

Proof of Theorem usgredg4
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgredg3.v . . . 4  |-  V  =  (Vtx `  G )
2 usgredg3.e . . . 4  |-  E  =  (iEdg `  G )
31, 2usgredg3 39457 . . 3  |-  ( ( G  e. USGraph  /\  X  e. 
dom  E )  ->  E. x  e.  V  E. z  e.  V  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) )
4 eleq2 2538 . . . . . . . 8  |-  ( ( E `  X )  =  { x ,  z }  ->  ( Y  e.  ( E `  X )  <->  Y  e.  { x ,  z } ) )
54adantl 473 . . . . . . 7  |-  ( ( x  =/=  z  /\  ( E `  X )  =  { x ,  z } )  -> 
( Y  e.  ( E `  X )  <-> 
Y  e.  { x ,  z } ) )
65adantl 473 . . . . . 6  |-  ( ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) )  ->  ( Y  e.  ( E `  X
)  <->  Y  e.  { x ,  z } ) )
7 elpri 3976 . . . . . . . 8  |-  ( Y  e.  { x ,  z }  ->  ( Y  =  x  \/  Y  =  z )
)
8 simplrr 779 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) )  ->  z  e.  V )
98adantl 473 . . . . . . . . . . . 12  |-  ( ( Y  =  x  /\  ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) ) )  ->  z  e.  V )
10 preq2 4043 . . . . . . . . . . . . . 14  |-  ( y  =  z  ->  { x ,  y }  =  { x ,  z } )
1110eqeq2d 2481 . . . . . . . . . . . . 13  |-  ( y  =  z  ->  ( { x ,  z }  =  { x ,  y }  <->  { x ,  z }  =  { x ,  z } ) )
1211adantl 473 . . . . . . . . . . . 12  |-  ( ( ( Y  =  x  /\  ( ( ( G  e. USGraph  /\  X  e. 
dom  E )  /\  ( x  e.  V  /\  z  e.  V
) )  /\  (
x  =/=  z  /\  ( E `  X )  =  { x ,  z } ) ) )  /\  y  =  z )  ->  ( { x ,  z }  =  { x ,  y }  <->  { x ,  z }  =  { x ,  z } ) )
13 eqidd 2472 . . . . . . . . . . . 12  |-  ( ( Y  =  x  /\  ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) ) )  ->  { x ,  z }  =  { x ,  z } )
149, 12, 13rspcedvd 3143 . . . . . . . . . . 11  |-  ( ( Y  =  x  /\  ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) ) )  ->  E. y  e.  V  { x ,  z }  =  { x ,  y } )
15 simprr 774 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) )  ->  ( E `  X )  =  {
x ,  z } )
16 preq1 4042 . . . . . . . . . . . . 13  |-  ( Y  =  x  ->  { Y ,  y }  =  { x ,  y } )
1715, 16eqeqan12rd 2489 . . . . . . . . . . . 12  |-  ( ( Y  =  x  /\  ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) ) )  ->  (
( E `  X
)  =  { Y ,  y }  <->  { x ,  z }  =  { x ,  y } ) )
1817rexbidv 2892 . . . . . . . . . . 11  |-  ( ( Y  =  x  /\  ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) ) )  ->  ( E. y  e.  V  ( E `  X )  =  { Y , 
y }  <->  E. y  e.  V  { x ,  z }  =  { x ,  y } ) )
1914, 18mpbird 240 . . . . . . . . . 10  |-  ( ( Y  =  x  /\  ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) ) )  ->  E. y  e.  V  ( E `  X )  =  { Y ,  y }
)
2019ex 441 . . . . . . . . 9  |-  ( Y  =  x  ->  (
( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) )  ->  E. y  e.  V  ( E `  X )  =  { Y ,  y }
) )
21 simplrl 778 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) )  ->  x  e.  V )
2221adantl 473 . . . . . . . . . . . 12  |-  ( ( Y  =  z  /\  ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) ) )  ->  x  e.  V )
23 preq2 4043 . . . . . . . . . . . . . 14  |-  ( y  =  x  ->  { z ,  y }  =  { z ,  x } )
2423eqeq2d 2481 . . . . . . . . . . . . 13  |-  ( y  =  x  ->  ( { x ,  z }  =  { z ,  y }  <->  { x ,  z }  =  { z ,  x } ) )
2524adantl 473 . . . . . . . . . . . 12  |-  ( ( ( Y  =  z  /\  ( ( ( G  e. USGraph  /\  X  e. 
dom  E )  /\  ( x  e.  V  /\  z  e.  V
) )  /\  (
x  =/=  z  /\  ( E `  X )  =  { x ,  z } ) ) )  /\  y  =  x )  ->  ( { x ,  z }  =  { z ,  y }  <->  { x ,  z }  =  { z ,  x } ) )
26 prcom 4041 . . . . . . . . . . . . 13  |-  { x ,  z }  =  { z ,  x }
2726a1i 11 . . . . . . . . . . . 12  |-  ( ( Y  =  z  /\  ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) ) )  ->  { x ,  z }  =  { z ,  x } )
2822, 25, 27rspcedvd 3143 . . . . . . . . . . 11  |-  ( ( Y  =  z  /\  ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) ) )  ->  E. y  e.  V  { x ,  z }  =  { z ,  y } )
29 preq1 4042 . . . . . . . . . . . . 13  |-  ( Y  =  z  ->  { Y ,  y }  =  { z ,  y } )
3015, 29eqeqan12rd 2489 . . . . . . . . . . . 12  |-  ( ( Y  =  z  /\  ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) ) )  ->  (
( E `  X
)  =  { Y ,  y }  <->  { x ,  z }  =  { z ,  y } ) )
3130rexbidv 2892 . . . . . . . . . . 11  |-  ( ( Y  =  z  /\  ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) ) )  ->  ( E. y  e.  V  ( E `  X )  =  { Y , 
y }  <->  E. y  e.  V  { x ,  z }  =  { z ,  y } ) )
3228, 31mpbird 240 . . . . . . . . . 10  |-  ( ( Y  =  z  /\  ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) ) )  ->  E. y  e.  V  ( E `  X )  =  { Y ,  y }
)
3332ex 441 . . . . . . . . 9  |-  ( Y  =  z  ->  (
( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) )  ->  E. y  e.  V  ( E `  X )  =  { Y ,  y }
) )
3420, 33jaoi 386 . . . . . . . 8  |-  ( ( Y  =  x  \/  Y  =  z )  ->  ( ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V
) )  /\  (
x  =/=  z  /\  ( E `  X )  =  { x ,  z } ) )  ->  E. y  e.  V  ( E `  X )  =  { Y , 
y } ) )
357, 34syl 17 . . . . . . 7  |-  ( Y  e.  { x ,  z }  ->  (
( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) )  ->  E. y  e.  V  ( E `  X )  =  { Y ,  y }
) )
3635com12 31 . . . . . 6  |-  ( ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) )  ->  ( Y  e.  { x ,  z }  ->  E. y  e.  V  ( E `  X )  =  { Y ,  y }
) )
376, 36sylbid 223 . . . . 5  |-  ( ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V ) )  /\  ( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } ) )  ->  ( Y  e.  ( E `  X
)  ->  E. y  e.  V  ( E `  X )  =  { Y ,  y }
) )
3837ex 441 . . . 4  |-  ( ( ( G  e. USGraph  /\  X  e.  dom  E )  /\  ( x  e.  V  /\  z  e.  V
) )  ->  (
( x  =/=  z  /\  ( E `  X
)  =  { x ,  z } )  ->  ( Y  e.  ( E `  X
)  ->  E. y  e.  V  ( E `  X )  =  { Y ,  y }
) ) )
3938rexlimdvva 2878 . . 3  |-  ( ( G  e. USGraph  /\  X  e. 
dom  E )  -> 
( E. x  e.  V  E. z  e.  V  ( x  =/=  z  /\  ( E `
 X )  =  { x ,  z } )  ->  ( Y  e.  ( E `  X )  ->  E. y  e.  V  ( E `  X )  =  { Y ,  y }
) ) )
403, 39mpd 15 . 2  |-  ( ( G  e. USGraph  /\  X  e. 
dom  E )  -> 
( Y  e.  ( E `  X )  ->  E. y  e.  V  ( E `  X )  =  { Y , 
y } ) )
41403impia 1228 1  |-  ( ( G  e. USGraph  /\  X  e. 
dom  E  /\  Y  e.  ( E `  X
) )  ->  E. y  e.  V  ( E `  X )  =  { Y ,  y }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757   {cpr 3961   dom cdm 4839   ` cfv 5589  Vtxcvtx 39251  iEdgciedg 39252   USGraph cusgr 39397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-umgr 39329  df-edga 39372  df-usgr 39399
This theorem is referenced by:  usgredgreu  39459
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