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

Theorem usgraedgprv 21388
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 21376 . 2  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( E `  X
)  C_  V )
2 usgraedg2 21387 . 2  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( # `  ( E `
 X ) )  =  2 )
3 sseq1 3361 . . . . 5  |-  ( ( E `  X )  =  { M ,  N }  ->  ( ( E `  X ) 
C_  V  <->  { M ,  N }  C_  V
) )
4 fveq2 5720 . . . . . 6  |-  ( ( E `  X )  =  { M ,  N }  ->  ( # `  ( E `  X
) )  =  (
# `  { M ,  N } ) )
54eqeq1d 2443 . . . . 5  |-  ( ( E `  X )  =  { M ,  N }  ->  ( (
# `  ( E `  X ) )  =  2  <->  ( # `  { M ,  N }
)  =  2 ) )
63, 5anbi12d 692 . . . 4  |-  ( ( E `  X )  =  { M ,  N }  ->  ( ( ( E `  X
)  C_  V  /\  ( # `  ( E `
 X ) )  =  2 )  <->  ( { M ,  N }  C_  V  /\  ( # `  { M ,  N } )  =  2 ) ) )
7 prssg 3945 . . . . . . 7  |-  ( ( M  e.  _V  /\  N  e.  _V )  ->  ( ( M  e.  V  /\  N  e.  V )  <->  { M ,  N }  C_  V
) )
87biimprd 215 . . . . . 6  |-  ( ( M  e.  _V  /\  N  e.  _V )  ->  ( { M ,  N }  C_  V  -> 
( M  e.  V  /\  N  e.  V
) ) )
98adantrd 455 . . . . 5  |-  ( ( M  e.  _V  /\  N  e.  _V )  ->  ( ( { M ,  N }  C_  V  /\  ( # `  { M ,  N }
)  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) ) )
10 ianor 475 . . . . . 6  |-  ( -.  ( M  e.  _V  /\  N  e.  _V )  <->  ( -.  M  e.  _V  \/  -.  N  e.  _V ) )
11 prprc1 3906 . . . . . . . . . . 11  |-  ( -.  M  e.  _V  ->  { M ,  N }  =  { N } )
12 fveq2 5720 . . . . . . . . . . . . . 14  |-  ( { M ,  N }  =  { N }  ->  (
# `  { M ,  N } )  =  ( # `  { N } ) )
13 hashsng 11639 . . . . . . . . . . . . . 14  |-  ( N  e.  _V  ->  ( # `
 { N }
)  =  1 )
1412, 13sylan9eqr 2489 . . . . . . . . . . . . 13  |-  ( ( N  e.  _V  /\  { M ,  N }  =  { N } )  ->  ( # `  { M ,  N }
)  =  1 )
15 eqtr2 2453 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { M ,  N }
)  =  1  /\  ( # `  { M ,  N }
)  =  2 )  ->  1  =  2 )
16 1ne2 10179 . . . . . . . . . . . . . . . . 17  |-  1  =/=  2
17 df-ne 2600 . . . . . . . . . . . . . . . . 17  |-  ( 1  =/=  2  <->  -.  1  =  2 )
1816, 17mpbi 200 . . . . . . . . . . . . . . . 16  |-  -.  1  =  2
1918pm2.21i 125 . . . . . . . . . . . . . . 15  |-  ( 1  =  2  ->  ( M  e.  V  /\  N  e.  V )
)
2015, 19syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( # `  { M ,  N }
)  =  1  /\  ( # `  { M ,  N }
)  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) )
2120ex 424 . . . . . . . . . . . . 13  |-  ( (
# `  { M ,  N } )  =  1  ->  ( ( # `
 { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
2214, 21syl 16 . . . . . . . . . . . 12  |-  ( ( N  e.  _V  /\  { M ,  N }  =  { N } )  ->  ( ( # `  { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
2322expcom 425 . . . . . . . . . . 11  |-  ( { M ,  N }  =  { N }  ->  ( N  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
2411, 23syl 16 . . . . . . . . . 10  |-  ( -.  M  e.  _V  ->  ( N  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
2524com12 29 . . . . . . . . 9  |-  ( N  e.  _V  ->  ( -.  M  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
26 prprc 3908 . . . . . . . . . . 11  |-  ( ( -.  N  e.  _V  /\ 
-.  M  e.  _V )  ->  { N ,  M }  =  (/) )
27 prcom 3874 . . . . . . . . . . . . 13  |-  { N ,  M }  =  { M ,  N }
2827eqeq1i 2442 . . . . . . . . . . . 12  |-  ( { N ,  M }  =  (/)  <->  { M ,  N }  =  (/) )
29 fveq2 5720 . . . . . . . . . . . . 13  |-  ( { M ,  N }  =  (/)  ->  ( # `  { M ,  N }
)  =  ( # `  (/) ) )
30 hash0 11638 . . . . . . . . . . . . 13  |-  ( # `  (/) )  =  0
3129, 30syl6eq 2483 . . . . . . . . . . . 12  |-  ( { M ,  N }  =  (/)  ->  ( # `  { M ,  N }
)  =  0 )
3228, 31sylbi 188 . . . . . . . . . . 11  |-  ( { N ,  M }  =  (/)  ->  ( # `  { M ,  N }
)  =  0 )
33 eqtr2 2453 . . . . . . . . . . . . 13  |-  ( ( ( # `  { M ,  N }
)  =  0  /\  ( # `  { M ,  N }
)  =  2 )  ->  0  =  2 )
34 2ne0 10075 . . . . . . . . . . . . . . . 16  |-  2  =/=  0
3534necomi 2680 . . . . . . . . . . . . . . 15  |-  0  =/=  2
36 df-ne 2600 . . . . . . . . . . . . . . 15  |-  ( 0  =/=  2  <->  -.  0  =  2 )
3735, 36mpbi 200 . . . . . . . . . . . . . 14  |-  -.  0  =  2
3837pm2.21i 125 . . . . . . . . . . . . 13  |-  ( 0  =  2  ->  ( M  e.  V  /\  N  e.  V )
)
3933, 38syl 16 . . . . . . . . . . . 12  |-  ( ( ( # `  { M ,  N }
)  =  0  /\  ( # `  { M ,  N }
)  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) )
4039ex 424 . . . . . . . . . . 11  |-  ( (
# `  { M ,  N } )  =  0  ->  ( ( # `
 { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
4126, 32, 403syl 19 . . . . . . . . . 10  |-  ( ( -.  N  e.  _V  /\ 
-.  M  e.  _V )  ->  ( ( # `  { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
4241ex 424 . . . . . . . . 9  |-  ( -.  N  e.  _V  ->  ( -.  M  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
4325, 42pm2.61i 158 . . . . . . . 8  |-  ( -.  M  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) )
44 prprc2 3907 . . . . . . . . . . 11  |-  ( -.  N  e.  _V  ->  { M ,  N }  =  { M } )
45 fveq2 5720 . . . . . . . . . . . . . 14  |-  ( { M ,  N }  =  { M }  ->  (
# `  { M ,  N } )  =  ( # `  { M } ) )
46 hashsng 11639 . . . . . . . . . . . . . 14  |-  ( M  e.  _V  ->  ( # `
 { M }
)  =  1 )
4745, 46sylan9eqr 2489 . . . . . . . . . . . . 13  |-  ( ( M  e.  _V  /\  { M ,  N }  =  { M } )  ->  ( # `  { M ,  N }
)  =  1 )
4847, 21syl 16 . . . . . . . . . . . 12  |-  ( ( M  e.  _V  /\  { M ,  N }  =  { M } )  ->  ( ( # `  { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
4948expcom 425 . . . . . . . . . . 11  |-  ( { M ,  N }  =  { M }  ->  ( M  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
5044, 49syl 16 . . . . . . . . . 10  |-  ( -.  N  e.  _V  ->  ( M  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
5150com12 29 . . . . . . . . 9  |-  ( M  e.  _V  ->  ( -.  N  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
52 prprc 3908 . . . . . . . . . . 11  |-  ( ( -.  M  e.  _V  /\ 
-.  N  e.  _V )  ->  { M ,  N }  =  (/) )
5352, 31, 403syl 19 . . . . . . . . . 10  |-  ( ( -.  M  e.  _V  /\ 
-.  N  e.  _V )  ->  ( ( # `  { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
5453ex 424 . . . . . . . . 9  |-  ( -.  M  e.  _V  ->  ( -.  N  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
5551, 54pm2.61i 158 . . . . . . . 8  |-  ( -.  N  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) )
5643, 55jaoi 369 . . . . . . 7  |-  ( ( -.  M  e.  _V  \/  -.  N  e.  _V )  ->  ( ( # `  { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
5756adantld 454 . . . . . 6  |-  ( ( -.  M  e.  _V  \/  -.  N  e.  _V )  ->  ( ( { M ,  N }  C_  V  /\  ( # `  { M ,  N } )  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) ) )
5810, 57sylbi 188 . . . . 5  |-  ( -.  ( M  e.  _V  /\  N  e.  _V )  ->  ( ( { M ,  N }  C_  V  /\  ( # `  { M ,  N }
)  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) ) )
599, 58pm2.61i 158 . . . 4  |-  ( ( { M ,  N }  C_  V  /\  ( # `
 { M ,  N } )  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) )
606, 59syl6bi 220 . . 3  |-  ( ( E `  X )  =  { M ,  N }  ->  ( ( ( E `  X
)  C_  V  /\  ( # `  ( E `
 X ) )  =  2 )  -> 
( M  e.  V  /\  N  e.  V
) ) )
6160com12 29 . 2  |-  ( ( ( E `  X
)  C_  V  /\  ( # `  ( E `
 X ) )  =  2 )  -> 
( ( E `  X )  =  { M ,  N }  ->  ( M  e.  V  /\  N  e.  V
) ) )
621, 2, 61syl2anc 643 1  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( ( E `  X )  =  { M ,  N }  ->  ( M  e.  V  /\  N  e.  V
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    C_ wss 3312   (/)c0 3620   {csn 3806   {cpr 3807   class class class wbr 4204   dom cdm 4870   ` cfv 5446   0cc0 8982   1c1 8983   2c2 10041   #chash 11610   USGrph cusg 21357
This theorem is referenced by:  usgranloopv  21390  usgranloop  21391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-hash 11611  df-usgra 21359
  Copyright terms: Public domain W3C validator