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Theorem usgraedgprv 23317
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 23305 . 2  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( E `  X
)  C_  V )
2 usgraedg2 23316 . 2  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( # `  ( E `
 X ) )  =  2 )
3 sseq1 3398 . . . . 5  |-  ( ( E `  X )  =  { M ,  N }  ->  ( ( E `  X ) 
C_  V  <->  { M ,  N }  C_  V
) )
4 fveq2 5712 . . . . . 6  |-  ( ( E `  X )  =  { M ,  N }  ->  ( # `  ( E `  X
) )  =  (
# `  { M ,  N } ) )
54eqeq1d 2451 . . . . 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 4049 . . . . . . 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 4006 . . . . . . . . . . 11  |-  ( -.  M  e.  _V  ->  { M ,  N }  =  { N } )
12 fveq2 5712 . . . . . . . . . . . . . 14  |-  ( { M ,  N }  =  { N }  ->  (
# `  { M ,  N } )  =  ( # `  { N } ) )
13 hashsng 12157 . . . . . . . . . . . . . 14  |-  ( N  e.  _V  ->  ( # `
 { N }
)  =  1 )
1412, 13sylan9eqr 2497 . . . . . . . . . . . . 13  |-  ( ( N  e.  _V  /\  { M ,  N }  =  { N } )  ->  ( # `  { M ,  N }
)  =  1 )
15 eqtr2 2461 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { M ,  N }
)  =  1  /\  ( # `  { M ,  N }
)  =  2 )  ->  1  =  2 )
16 1ne2 10555 . . . . . . . . . . . . . . . . 17  |-  1  =/=  2
1716neii 2624 . . . . . . . . . . . . . . . 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 4008 . . . . . . . . . . 11  |-  ( ( -.  N  e.  _V  /\ 
-.  M  e.  _V )  ->  { N ,  M }  =  (/) )
26 prcom 3974 . . . . . . . . . . . . 13  |-  { N ,  M }  =  { M ,  N }
2726eqeq1i 2450 . . . . . . . . . . . 12  |-  ( { N ,  M }  =  (/)  <->  { M ,  N }  =  (/) )
28 fveq2 5712 . . . . . . . . . . . . 13  |-  ( { M ,  N }  =  (/)  ->  ( # `  { M ,  N }
)  =  ( # `  (/) ) )
29 hash0 12156 . . . . . . . . . . . . 13  |-  ( # `  (/) )  =  0
3028, 29syl6eq 2491 . . . . . . . . . . . 12  |-  ( { M ,  N }  =  (/)  ->  ( # `  { M ,  N }
)  =  0 )
3127, 30sylbi 195 . . . . . . . . . . 11  |-  ( { N ,  M }  =  (/)  ->  ( # `  { M ,  N }
)  =  0 )
32 eqtr2 2461 . . . . . . . . . . . . 13  |-  ( ( ( # `  { M ,  N }
)  =  0  /\  ( # `  { M ,  N }
)  =  2 )  ->  0  =  2 )
33 2ne0 10435 . . . . . . . . . . . . . . 15  |-  2  =/=  0
3433nesymi 2672 . . . . . . . . . . . . . 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 4007 . . . . . . . . . . 11  |-  ( -.  N  e.  _V  ->  { M ,  N }  =  { M } )
42 fveq2 5712 . . . . . . . . . . . . . 14  |-  ( { M ,  N }  =  { M }  ->  (
# `  { M ,  N } )  =  ( # `  { M } ) )
43 hashsng 12157 . . . . . . . . . . . . . 14  |-  ( M  e.  _V  ->  ( # `
 { M }
)  =  1 )
4442, 43sylan9eqr 2497 . . . . . . . . . . . . 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 4008 . . . . . . . . . . 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 1369    e. wcel 1756   _Vcvv 2993    C_ wss 3349   (/)c0 3658   {csn 3898   {cpr 3900   class class class wbr 4313   dom cdm 4861   ` cfv 5439   0cc0 9303   1c1 9304   2c2 10392   #chash 12124   USGrph cusg 23286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-hash 12125  df-usgra 23288
This theorem is referenced by:  usgranloopv  23319  usgranloop  23320
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