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Theorem eupai 21679
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 5536 . . . . 5  |-  ( E  Fn  A  ->  dom  E  =  A )
2 iseupa 21677 . . . . 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 16 . . . 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 473 . . 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 450 . 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 5667 . . . . . . . . . . . . . 14  |-  ( F : ( 1 ... n ) -1-1-onto-> A  ->  F  Fn  ( 1 ... n
) )
76ad2antll 710 . . . . . . . . . . . . 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 11302 . . . . . . . . . . . . 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 7175 . . . . . . . . . . . . 13  |-  ( ( F  Fn  ( 1 ... n )  /\  ( 1 ... n
)  e.  Fin )  ->  ( 1 ... n
)  ~~  F )
107, 8, 9syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( 1 ... n )  ~~  F )
11 enfi 7317 . . . . . . . . . . . . . . 15  |-  ( ( 1 ... n ) 
~~  F  ->  (
( 1 ... n
)  e.  Fin  <->  F  e.  Fin ) )
1210, 11syl 16 . . . . . . . . . . . . . 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 202 . . . . . . . . . . . . 13  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  F  e.  Fin )
14 hashen 11621 . . . . . . . . . . . . 13  |-  ( ( ( 1 ... n
)  e.  Fin  /\  F  e.  Fin )  ->  ( ( # `  (
1 ... n ) )  =  ( # `  F
)  <->  ( 1 ... n )  ~~  F
) )
158, 13, 14syl2anc 643 . . . . . . . . . . . 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 224 . . . . . . . . . . 11  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  (
1 ... n ) )  =  ( # `  F
) )
17 hashfz1 11620 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  ( # `  ( 1 ... n
) )  =  n )
1817ad2antrl 709 . . . . . . . . . . 11  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  (
1 ... n ) )  =  n )
1916, 18eqtr3d 2469 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  F
)  =  n )
20 simprl 733 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  n  e.  NN0 )
2119, 20eqeltrd 2509 . . . . . . . . 9  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  F
)  e.  NN0 )
2221a1d 23 . . . . . . . 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 734 . . . . . . . . . 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 6089 . . . . . . . . . . 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 5658 . . . . . . . . . . 11  |-  ( ( 1 ... ( # `  F ) )  =  ( 1 ... n
)  ->  ( F : ( 1 ... ( # `  F
) ) -1-1-onto-> A  <->  F : ( 1 ... n ) -1-1-onto-> A ) )
2624, 25syl 16 . . . . . . . . . 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 224 . . . . . . . . 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 23 . . . . . . . 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 6089 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( 0 ... ( # `  F
) )  =  ( 0 ... n ) )
3029feq2d 5573 . . . . . . . . 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 215 . . . . . . . 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 1135 . . . . . . 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 2902 . . . . . . . 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 215 . . . . . . 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 547 . . . . . 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 ) } ) ) )
3635exp3a 426 . . . . 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 599 . . . 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 1167 . . 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 2822 . 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   {cpr 3807   class class class wbr 4204   dom cdm 4870    Fn wfn 5441   -->wf 5442   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073    ~~ cen 7098   Fincfn 7101   0cc0 8980   1c1 8981    - cmin 9281   NN0cn0 10211   ...cfz 11033   #chash 11608   UMGrph cumg 21337   EulPaths ceup 21674
This theorem is referenced by:  eupacl  21681  eupaf1o  21682  eupapf  21684  eupaseg  21685
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-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
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-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7816  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-hash 11609  df-umgra 21338  df-eupa 21675
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