MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgraedgprv Structured version   Unicode version

Theorem usgraedgprv 24038
Description: In an undirected graph, an edge is an unordered pair of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
Assertion
Ref Expression
usgraedgprv  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( ( E `  X )  =  { M ,  N }  ->  ( M  e.  V  /\  N  e.  V
) ) )

Proof of Theorem usgraedgprv
StepHypRef Expression
1 usgrass 24024 . 2  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( E `  X
)  C_  V )
2 usgraedg2 24037 . 2  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( # `  ( E `
 X ) )  =  2 )
3 sseq1 3518 . . . . 5  |-  ( ( E `  X )  =  { M ,  N }  ->  ( ( E `  X ) 
C_  V  <->  { M ,  N }  C_  V
) )
4 fveq2 5857 . . . . . 6  |-  ( ( E `  X )  =  { M ,  N }  ->  ( # `  ( E `  X
) )  =  (
# `  { M ,  N } ) )
54eqeq1d 2462 . . . . 5  |-  ( ( E `  X )  =  { M ,  N }  ->  ( (
# `  ( E `  X ) )  =  2  <->  ( # `  { M ,  N }
)  =  2 ) )
63, 5anbi12d 710 . . . 4  |-  ( ( E `  X )  =  { M ,  N }  ->  ( ( ( E `  X
)  C_  V  /\  ( # `  ( E `
 X ) )  =  2 )  <->  ( { M ,  N }  C_  V  /\  ( # `  { M ,  N } )  =  2 ) ) )
7 prssg 4175 . . . . . . 7  |-  ( ( M  e.  _V  /\  N  e.  _V )  ->  ( ( M  e.  V  /\  N  e.  V )  <->  { M ,  N }  C_  V
) )
87biimprd 223 . . . . . 6  |-  ( ( M  e.  _V  /\  N  e.  _V )  ->  ( { M ,  N }  C_  V  -> 
( M  e.  V  /\  N  e.  V
) ) )
98adantrd 468 . . . . 5  |-  ( ( M  e.  _V  /\  N  e.  _V )  ->  ( ( { M ,  N }  C_  V  /\  ( # `  { M ,  N }
)  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) ) )
10 ianor 488 . . . . . 6  |-  ( -.  ( M  e.  _V  /\  N  e.  _V )  <->  ( -.  M  e.  _V  \/  -.  N  e.  _V ) )
11 prprc1 4130 . . . . . . . . . . 11  |-  ( -.  M  e.  _V  ->  { M ,  N }  =  { N } )
12 fveq2 5857 . . . . . . . . . . . . . 14  |-  ( { M ,  N }  =  { N }  ->  (
# `  { M ,  N } )  =  ( # `  { N } ) )
13 hashsng 12393 . . . . . . . . . . . . . 14  |-  ( N  e.  _V  ->  ( # `
 { N }
)  =  1 )
1412, 13sylan9eqr 2523 . . . . . . . . . . . . 13  |-  ( ( N  e.  _V  /\  { M ,  N }  =  { N } )  ->  ( # `  { M ,  N }
)  =  1 )
15 eqtr2 2487 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { M ,  N }
)  =  1  /\  ( # `  { M ,  N }
)  =  2 )  ->  1  =  2 )
16 1ne2 10737 . . . . . . . . . . . . . . . . 17  |-  1  =/=  2
1716neii 2659 . . . . . . . . . . . . . . . 16  |-  -.  1  =  2
1817pm2.21i 131 . . . . . . . . . . . . . . 15  |-  ( 1  =  2  ->  ( M  e.  V  /\  N  e.  V )
)
1915, 18syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( # `  { M ,  N }
)  =  1  /\  ( # `  { M ,  N }
)  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) )
2019ex 434 . . . . . . . . . . . . 13  |-  ( (
# `  { M ,  N } )  =  1  ->  ( ( # `
 { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
2114, 20syl 16 . . . . . . . . . . . 12  |-  ( ( N  e.  _V  /\  { M ,  N }  =  { N } )  ->  ( ( # `  { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
2221expcom 435 . . . . . . . . . . 11  |-  ( { M ,  N }  =  { N }  ->  ( N  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
2311, 22syl 16 . . . . . . . . . 10  |-  ( -.  M  e.  _V  ->  ( N  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
2423com12 31 . . . . . . . . 9  |-  ( N  e.  _V  ->  ( -.  M  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
25 prprc 4132 . . . . . . . . . . 11  |-  ( ( -.  N  e.  _V  /\ 
-.  M  e.  _V )  ->  { N ,  M }  =  (/) )
26 prcom 4098 . . . . . . . . . . . . 13  |-  { N ,  M }  =  { M ,  N }
2726eqeq1i 2467 . . . . . . . . . . . 12  |-  ( { N ,  M }  =  (/)  <->  { M ,  N }  =  (/) )
28 fveq2 5857 . . . . . . . . . . . . 13  |-  ( { M ,  N }  =  (/)  ->  ( # `  { M ,  N }
)  =  ( # `  (/) ) )
29 hash0 12392 . . . . . . . . . . . . 13  |-  ( # `  (/) )  =  0
3028, 29syl6eq 2517 . . . . . . . . . . . 12  |-  ( { M ,  N }  =  (/)  ->  ( # `  { M ,  N }
)  =  0 )
3127, 30sylbi 195 . . . . . . . . . . 11  |-  ( { N ,  M }  =  (/)  ->  ( # `  { M ,  N }
)  =  0 )
32 eqtr2 2487 . . . . . . . . . . . . 13  |-  ( ( ( # `  { M ,  N }
)  =  0  /\  ( # `  { M ,  N }
)  =  2 )  ->  0  =  2 )
33 2ne0 10617 . . . . . . . . . . . . . . 15  |-  2  =/=  0
3433nesymi 2733 . . . . . . . . . . . . . 14  |-  -.  0  =  2
3534pm2.21i 131 . . . . . . . . . . . . 13  |-  ( 0  =  2  ->  ( M  e.  V  /\  N  e.  V )
)
3632, 35syl 16 . . . . . . . . . . . 12  |-  ( ( ( # `  { M ,  N }
)  =  0  /\  ( # `  { M ,  N }
)  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) )
3736ex 434 . . . . . . . . . . 11  |-  ( (
# `  { M ,  N } )  =  0  ->  ( ( # `
 { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
3825, 31, 373syl 20 . . . . . . . . . 10  |-  ( ( -.  N  e.  _V  /\ 
-.  M  e.  _V )  ->  ( ( # `  { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
3938ex 434 . . . . . . . . 9  |-  ( -.  N  e.  _V  ->  ( -.  M  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
4024, 39pm2.61i 164 . . . . . . . 8  |-  ( -.  M  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) )
41 prprc2 4131 . . . . . . . . . . 11  |-  ( -.  N  e.  _V  ->  { M ,  N }  =  { M } )
42 fveq2 5857 . . . . . . . . . . . . . 14  |-  ( { M ,  N }  =  { M }  ->  (
# `  { M ,  N } )  =  ( # `  { M } ) )
43 hashsng 12393 . . . . . . . . . . . . . 14  |-  ( M  e.  _V  ->  ( # `
 { M }
)  =  1 )
4442, 43sylan9eqr 2523 . . . . . . . . . . . . 13  |-  ( ( M  e.  _V  /\  { M ,  N }  =  { M } )  ->  ( # `  { M ,  N }
)  =  1 )
4544, 20syl 16 . . . . . . . . . . . 12  |-  ( ( M  e.  _V  /\  { M ,  N }  =  { M } )  ->  ( ( # `  { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
4645expcom 435 . . . . . . . . . . 11  |-  ( { M ,  N }  =  { M }  ->  ( M  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
4741, 46syl 16 . . . . . . . . . 10  |-  ( -.  N  e.  _V  ->  ( M  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
4847com12 31 . . . . . . . . 9  |-  ( M  e.  _V  ->  ( -.  N  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
49 prprc 4132 . . . . . . . . . . 11  |-  ( ( -.  M  e.  _V  /\ 
-.  N  e.  _V )  ->  { M ,  N }  =  (/) )
5049, 30, 373syl 20 . . . . . . . . . 10  |-  ( ( -.  M  e.  _V  /\ 
-.  N  e.  _V )  ->  ( ( # `  { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
5150ex 434 . . . . . . . . 9  |-  ( -.  M  e.  _V  ->  ( -.  N  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
5248, 51pm2.61i 164 . . . . . . . 8  |-  ( -.  N  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) )
5340, 52jaoi 379 . . . . . . 7  |-  ( ( -.  M  e.  _V  \/  -.  N  e.  _V )  ->  ( ( # `  { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
5453adantld 467 . . . . . 6  |-  ( ( -.  M  e.  _V  \/  -.  N  e.  _V )  ->  ( ( { M ,  N }  C_  V  /\  ( # `  { M ,  N } )  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) ) )
5510, 54sylbi 195 . . . . 5  |-  ( -.  ( M  e.  _V  /\  N  e.  _V )  ->  ( ( { M ,  N }  C_  V  /\  ( # `  { M ,  N }
)  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) ) )
569, 55pm2.61i 164 . . . 4  |-  ( ( { M ,  N }  C_  V  /\  ( # `
 { M ,  N } )  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) )
576, 56syl6bi 228 . . 3  |-  ( ( E `  X )  =  { M ,  N }  ->  ( ( ( E `  X
)  C_  V  /\  ( # `  ( E `
 X ) )  =  2 )  -> 
( M  e.  V  /\  N  e.  V
) ) )
5857com12 31 . 2  |-  ( ( ( E `  X
)  C_  V  /\  ( # `  ( E `
 X ) )  =  2 )  -> 
( ( E `  X )  =  { M ,  N }  ->  ( M  e.  V  /\  N  e.  V
) ) )
591, 2, 58syl2anc 661 1  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( ( E `  X )  =  { M ,  N }  ->  ( M  e.  V  /\  N  e.  V
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106    C_ wss 3469   (/)c0 3778   {csn 4020   {cpr 4022   class class class wbr 4440   dom cdm 4992   ` cfv 5579   0cc0 9481   1c1 9482   2c2 10574   #chash 12360   USGrph cusg 23993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-hash 12361  df-usgra 23996
This theorem is referenced by:  usgranloopv  24040  usgranloop  24041
  Copyright terms: Public domain W3C validator