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Theorem usgraedgprv 24674
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 24659 . 2  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( E `  X
)  C_  V )
2 usgraedg2 24673 . 2  |-  ( ( V USGrph  E  /\  X  e. 
dom  E )  -> 
( # `  ( E `
 X ) )  =  2 )
3 sseq1 3462 . . . . 5  |-  ( ( E `  X )  =  { M ,  N }  ->  ( ( E `  X ) 
C_  V  <->  { M ,  N }  C_  V
) )
4 fveq2 5805 . . . . . 6  |-  ( ( E `  X )  =  { M ,  N }  ->  ( # `  ( E `  X
) )  =  (
# `  { M ,  N } ) )
54eqeq1d 2404 . . . . 5  |-  ( ( E `  X )  =  { M ,  N }  ->  ( (
# `  ( E `  X ) )  =  2  <->  ( # `  { M ,  N }
)  =  2 ) )
63, 5anbi12d 709 . . . 4  |-  ( ( E `  X )  =  { M ,  N }  ->  ( ( ( E `  X
)  C_  V  /\  ( # `  ( E `
 X ) )  =  2 )  <->  ( { M ,  N }  C_  V  /\  ( # `  { M ,  N } )  =  2 ) ) )
7 prssg 4126 . . . . . . 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 466 . . . . 5  |-  ( ( M  e.  _V  /\  N  e.  _V )  ->  ( ( { M ,  N }  C_  V  /\  ( # `  { M ,  N }
)  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) ) )
10 ianor 486 . . . . . 6  |-  ( -.  ( M  e.  _V  /\  N  e.  _V )  <->  ( -.  M  e.  _V  \/  -.  N  e.  _V ) )
11 prprc1 4081 . . . . . . . . . . 11  |-  ( -.  M  e.  _V  ->  { M ,  N }  =  { N } )
12 fveq2 5805 . . . . . . . . . . . . . 14  |-  ( { M ,  N }  =  { N }  ->  (
# `  { M ,  N } )  =  ( # `  { N } ) )
13 hashsng 12393 . . . . . . . . . . . . . 14  |-  ( N  e.  _V  ->  ( # `
 { N }
)  =  1 )
1412, 13sylan9eqr 2465 . . . . . . . . . . . . 13  |-  ( ( N  e.  _V  /\  { M ,  N }  =  { N } )  ->  ( # `  { M ,  N }
)  =  1 )
15 eqtr2 2429 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { M ,  N }
)  =  1  /\  ( # `  { M ,  N }
)  =  2 )  ->  1  =  2 )
16 1ne2 10709 . . . . . . . . . . . . . . . . 17  |-  1  =/=  2
1716neii 2602 . . . . . . . . . . . . . . . 16  |-  -.  1  =  2
1817pm2.21i 131 . . . . . . . . . . . . . . 15  |-  ( 1  =  2  ->  ( M  e.  V  /\  N  e.  V )
)
1915, 18syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( # `  { M ,  N }
)  =  1  /\  ( # `  { M ,  N }
)  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) )
2019ex 432 . . . . . . . . . . . . 13  |-  ( (
# `  { M ,  N } )  =  1  ->  ( ( # `
 { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
2114, 20syl 17 . . . . . . . . . . . 12  |-  ( ( N  e.  _V  /\  { M ,  N }  =  { N } )  ->  ( ( # `  { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
2221expcom 433 . . . . . . . . . . 11  |-  ( { M ,  N }  =  { N }  ->  ( N  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
2311, 22syl 17 . . . . . . . . . 10  |-  ( -.  M  e.  _V  ->  ( N  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
2423com12 29 . . . . . . . . 9  |-  ( N  e.  _V  ->  ( -.  M  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
25 prprc 4083 . . . . . . . . . . 11  |-  ( ( -.  N  e.  _V  /\ 
-.  M  e.  _V )  ->  { N ,  M }  =  (/) )
26 prcom 4049 . . . . . . . . . . . . 13  |-  { N ,  M }  =  { M ,  N }
2726eqeq1i 2409 . . . . . . . . . . . 12  |-  ( { N ,  M }  =  (/)  <->  { M ,  N }  =  (/) )
28 fveq2 5805 . . . . . . . . . . . . 13  |-  ( { M ,  N }  =  (/)  ->  ( # `  { M ,  N }
)  =  ( # `  (/) ) )
29 hash0 12392 . . . . . . . . . . . . 13  |-  ( # `  (/) )  =  0
3028, 29syl6eq 2459 . . . . . . . . . . . 12  |-  ( { M ,  N }  =  (/)  ->  ( # `  { M ,  N }
)  =  0 )
3127, 30sylbi 195 . . . . . . . . . . 11  |-  ( { N ,  M }  =  (/)  ->  ( # `  { M ,  N }
)  =  0 )
32 eqtr2 2429 . . . . . . . . . . . . 13  |-  ( ( ( # `  { M ,  N }
)  =  0  /\  ( # `  { M ,  N }
)  =  2 )  ->  0  =  2 )
33 2ne0 10589 . . . . . . . . . . . . . . 15  |-  2  =/=  0
3433nesymi 2676 . . . . . . . . . . . . . 14  |-  -.  0  =  2
3534pm2.21i 131 . . . . . . . . . . . . 13  |-  ( 0  =  2  ->  ( M  e.  V  /\  N  e.  V )
)
3632, 35syl 17 . . . . . . . . . . . 12  |-  ( ( ( # `  { M ,  N }
)  =  0  /\  ( # `  { M ,  N }
)  =  2 )  ->  ( M  e.  V  /\  N  e.  V ) )
3736ex 432 . . . . . . . . . . 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 432 . . . . . . . . 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 4082 . . . . . . . . . . 11  |-  ( -.  N  e.  _V  ->  { M ,  N }  =  { M } )
42 fveq2 5805 . . . . . . . . . . . . . 14  |-  ( { M ,  N }  =  { M }  ->  (
# `  { M ,  N } )  =  ( # `  { M } ) )
43 hashsng 12393 . . . . . . . . . . . . . 14  |-  ( M  e.  _V  ->  ( # `
 { M }
)  =  1 )
4442, 43sylan9eqr 2465 . . . . . . . . . . . . 13  |-  ( ( M  e.  _V  /\  { M ,  N }  =  { M } )  ->  ( # `  { M ,  N }
)  =  1 )
4544, 20syl 17 . . . . . . . . . . . 12  |-  ( ( M  e.  _V  /\  { M ,  N }  =  { M } )  ->  ( ( # `  { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
4645expcom 433 . . . . . . . . . . 11  |-  ( { M ,  N }  =  { M }  ->  ( M  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
4741, 46syl 17 . . . . . . . . . 10  |-  ( -.  N  e.  _V  ->  ( M  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
4847com12 29 . . . . . . . . 9  |-  ( M  e.  _V  ->  ( -.  N  e.  _V  ->  ( ( # `  { M ,  N }
)  =  2  -> 
( M  e.  V  /\  N  e.  V
) ) ) )
49 prprc 4083 . . . . . . . . . . 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 432 . . . . . . . . 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 377 . . . . . . 7  |-  ( ( -.  M  e.  _V  \/  -.  N  e.  _V )  ->  ( ( # `  { M ,  N } )  =  2  ->  ( M  e.  V  /\  N  e.  V ) ) )
5453adantld 465 . . . . . 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 29 . 2  |-  ( ( ( E `  X
)  C_  V  /\  ( # `  ( E `
 X ) )  =  2 )  -> 
( ( E `  X )  =  { M ,  N }  ->  ( M  e.  V  /\  N  e.  V
) ) )
591, 2, 58syl2anc 659 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 366    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058    C_ wss 3413   (/)c0 3737   {csn 3971   {cpr 3973   class class class wbr 4394   dom cdm 4942   ` cfv 5525   0cc0 9442   1c1 9443   2c2 10546   #chash 12359   USGrph cusg 24628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-card 8272  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-n0 10757  df-z 10826  df-uz 11046  df-fz 11644  df-hash 12360  df-usgra 24631
This theorem is referenced by:  usgranloopv  24676  usgranloop  24677
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