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Theorem eupai 25540
Description: Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.)
Assertion
Ref Expression
eupai  |-  ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  (
( ( # `  F
)  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  /\  P :
( 0 ... ( # `
 F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) )
Distinct variable groups:    A, k    k, E    k, F    P, k    k, V

Proof of Theorem eupai
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 fndm 5693 . . . . 5  |-  ( E  Fn  A  ->  dom  E  =  A )
2 iseupa 25538 . . . . 5  |-  ( dom 
E  =  A  -> 
( F ( V EulPaths  E ) P  <->  ( V UMGrph  E  /\  E. n  e. 
NN0  ( F :
( 1 ... n
)
-1-1-onto-> A  /\  P : ( 0 ... n ) --> V  /\  A. k  e.  ( 1 ... n
) ( E `  ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) ) ) )
31, 2syl 17 . . . 4  |-  ( E  Fn  A  ->  ( F ( V EulPaths  E ) P  <->  ( V UMGrph  E  /\  E. n  e.  NN0  ( F : ( 1 ... n ) -1-1-onto-> A  /\  P : ( 0 ... n ) --> V  /\  A. k  e.  ( 1 ... n ) ( E `  ( F `
 k ) )  =  { ( P `
 ( k  - 
1 ) ) ,  ( P `  k
) } ) ) ) )
43biimpac 488 . . 3  |-  ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  ( V UMGrph  E  /\  E. n  e.  NN0  ( F :
( 1 ... n
)
-1-1-onto-> A  /\  P : ( 0 ... n ) --> V  /\  A. k  e.  ( 1 ... n
) ( E `  ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) ) )
54simprd 464 . 2  |-  ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  E. n  e.  NN0  ( F :
( 1 ... n
)
-1-1-onto-> A  /\  P : ( 0 ... n ) --> V  /\  A. k  e.  ( 1 ... n
) ( E `  ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) )
6 f1ofn 5832 . . . . . . . . . . . . . 14  |-  ( F : ( 1 ... n ) -1-1-onto-> A  ->  F  Fn  ( 1 ... n
) )
76ad2antll 733 . . . . . . . . . . . . 13  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  F  Fn  ( 1 ... n
) )
8 fzfid 12183 . . . . . . . . . . . . 13  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( 1 ... n )  e. 
Fin )
9 fndmeng 7653 . . . . . . . . . . . . 13  |-  ( ( F  Fn  ( 1 ... n )  /\  ( 1 ... n
)  e.  Fin )  ->  ( 1 ... n
)  ~~  F )
107, 8, 9syl2anc 665 . . . . . . . . . . . 12  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( 1 ... n )  ~~  F )
11 enfi 7794 . . . . . . . . . . . . . . 15  |-  ( ( 1 ... n ) 
~~  F  ->  (
( 1 ... n
)  e.  Fin  <->  F  e.  Fin ) )
1210, 11syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( (
1 ... n )  e. 
Fin 
<->  F  e.  Fin )
)
138, 12mpbid 213 . . . . . . . . . . . . 13  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  F  e.  Fin )
14 hashen 12527 . . . . . . . . . . . . 13  |-  ( ( ( 1 ... n
)  e.  Fin  /\  F  e.  Fin )  ->  ( ( # `  (
1 ... n ) )  =  ( # `  F
)  <->  ( 1 ... n )  ~~  F
) )
158, 13, 14syl2anc 665 . . . . . . . . . . . 12  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( ( # `
 ( 1 ... n ) )  =  ( # `  F
)  <->  ( 1 ... n )  ~~  F
) )
1610, 15mpbird 235 . . . . . . . . . . 11  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  (
1 ... n ) )  =  ( # `  F
) )
17 hashfz1 12526 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  ( # `  ( 1 ... n
) )  =  n )
1817ad2antrl 732 . . . . . . . . . . 11  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  (
1 ... n ) )  =  n )
1916, 18eqtr3d 2472 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  F
)  =  n )
20 simprl 762 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  n  e.  NN0 )
2119, 20eqeltrd 2517 . . . . . . . . 9  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  F
)  e.  NN0 )
2221a1d 26 . . . . . . . 8  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( P : ( 0 ... n ) --> V  -> 
( # `  F )  e.  NN0 ) )
23 simprr 764 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  F :
( 1 ... n
)
-1-1-onto-> A )
2419oveq2d 6321 . . . . . . . . . . 11  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( 1 ... ( # `  F
) )  =  ( 1 ... n ) )
25 f1oeq2 5823 . . . . . . . . . . 11  |-  ( ( 1 ... ( # `  F ) )  =  ( 1 ... n
)  ->  ( F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  <->  F : ( 1 ... n ) -1-1-onto-> A ) )
2624, 25syl 17 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  <->  F : ( 1 ... n ) -1-1-onto-> A ) )
2723, 26mpbird 235 . . . . . . . . 9  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  F :
( 1 ... ( # `
 F ) ) -1-1-onto-> A )
2827a1d 26 . . . . . . . 8  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( P : ( 0 ... n ) --> V  ->  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A ) )
2919oveq2d 6321 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( 0 ... ( # `  F
) )  =  ( 0 ... n ) )
3029feq2d 5733 . . . . . . . . 9  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( P : ( 0 ... ( # `  F
) ) --> V  <->  P :
( 0 ... n
) --> V ) )
3130biimprd 226 . . . . . . . 8  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( P : ( 0 ... n ) --> V  ->  P : ( 0 ... ( # `  F
) ) --> V ) )
3222, 28, 313jcad 1186 . . . . . . 7  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( P : ( 0 ... n ) --> V  -> 
( ( # `  F
)  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  /\  P :
( 0 ... ( # `
 F ) ) --> V ) ) )
3324raleqdv 3038 . . . . . . . 8  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( A. k  e.  ( 1 ... ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) }  <->  A. k  e.  ( 1 ... n ) ( E `  ( F `  k )
)  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) )
3433biimprd 226 . . . . . . 7  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( A. k  e.  ( 1 ... n ) ( E `  ( F `
 k ) )  =  { ( P `
 ( k  - 
1 ) ) ,  ( P `  k
) }  ->  A. k  e.  ( 1 ... ( # `
 F ) ) ( E `  ( F `  k )
)  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) )
3532, 34anim12d 565 . . . . . 6  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( ( P : ( 0 ... n ) --> V  /\  A. k  e.  ( 1 ... n ) ( E `  ( F `
 k ) )  =  { ( P `
 ( k  - 
1 ) ) ,  ( P `  k
) } )  -> 
( ( ( # `  F )  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  /\  P :
( 0 ... ( # `
 F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) ) )
3635expd 437 . . . . 5  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( P : ( 0 ... n ) --> V  -> 
( A. k  e.  ( 1 ... n
) ( E `  ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) }  ->  ( ( ( # `  F
)  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  /\  P :
( 0 ... ( # `
 F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) ) ) )
3736expr 618 . . . 4  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  n  e.  NN0 )  ->  ( F :
( 1 ... n
)
-1-1-onto-> A  ->  ( P :
( 0 ... n
) --> V  ->  ( A. k  e.  (
1 ... n ) ( E `  ( F `
 k ) )  =  { ( P `
 ( k  - 
1 ) ) ,  ( P `  k
) }  ->  (
( ( # `  F
)  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  /\  P :
( 0 ... ( # `
 F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) ) ) ) )
38373impd 1219 . . 3  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  n  e.  NN0 )  ->  ( ( F : ( 1 ... n ) -1-1-onto-> A  /\  P :
( 0 ... n
) --> V  /\  A. k  e.  ( 1 ... n ) ( E `  ( F `
 k ) )  =  { ( P `
 ( k  - 
1 ) ) ,  ( P `  k
) } )  -> 
( ( ( # `  F )  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  /\  P :
( 0 ... ( # `
 F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) ) )
3938rexlimdva 2924 . 2  |-  ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  ( E. n  e.  NN0  ( F : ( 1 ... n ) -1-1-onto-> A  /\  P : ( 0 ... n ) --> V  /\  A. k  e.  ( 1 ... n ) ( E `  ( F `
 k ) )  =  { ( P `
 ( k  - 
1 ) ) ,  ( P `  k
) } )  -> 
( ( ( # `  F )  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  /\  P :
( 0 ... ( # `
 F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) ) )
405, 39mpd 15 1  |-  ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  (
( ( # `  F
)  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  /\  P :
( 0 ... ( # `
 F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  ( k  -  1 ) ) ,  ( P `  k ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783   {cpr 4004   class class class wbr 4426   dom cdm 4854    Fn wfn 5596   -->wf 5597   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305    ~~ cen 7574   Fincfn 7577   0cc0 9538   1c1 9539    - cmin 9859   NN0cn0 10869   ...cfz 11782   #chash 12512   UMGrph cumg 24885   EulPaths ceup 25535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-er 7371  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-hash 12513  df-umgra 24886  df-eupa 25536
This theorem is referenced by:  eupacl  25542  eupaf1o  25543  eupapf  25545  eupaseg  25546
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