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Theorem eupai 24671
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 5680 . . . . 5  |-  ( E  Fn  A  ->  dom  E  =  A )
2 iseupa 24669 . . . . 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 486 . . 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 463 . 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 5817 . . . . . . . . . . . . . 14  |-  ( F : ( 1 ... n ) -1-1-onto-> A  ->  F  Fn  ( 1 ... n
) )
76ad2antll 728 . . . . . . . . . . . . 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 12051 . . . . . . . . . . . . 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 7592 . . . . . . . . . . . . 13  |-  ( ( F  Fn  ( 1 ... n )  /\  ( 1 ... n
)  e.  Fin )  ->  ( 1 ... n
)  ~~  F )
107, 8, 9syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( 1 ... n )  ~~  F )
11 enfi 7736 . . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . 13  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  F  e.  Fin )
14 hashen 12388 . . . . . . . . . . . . 13  |-  ( ( ( 1 ... n
)  e.  Fin  /\  F  e.  Fin )  ->  ( ( # `  (
1 ... n ) )  =  ( # `  F
)  <->  ( 1 ... n )  ~~  F
) )
158, 13, 14syl2anc 661 . . . . . . . . . . . 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 232 . . . . . . . . . . 11  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  (
1 ... n ) )  =  ( # `  F
) )
17 hashfz1 12387 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  ( # `  ( 1 ... n
) )  =  n )
1817ad2antrl 727 . . . . . . . . . . 11  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  (
1 ... n ) )  =  n )
1916, 18eqtr3d 2510 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  F
)  =  n )
20 simprl 755 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  n  e.  NN0 )
2119, 20eqeltrd 2555 . . . . . . . . 9  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( # `  F
)  e.  NN0 )
2221a1d 25 . . . . . . . 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 756 . . . . . . . . . 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 6300 . . . . . . . . . . 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 5808 . . . . . . . . . . 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 232 . . . . . . . . 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 25 . . . . . . . 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 6300 . . . . . . . . . 10  |-  ( ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  /\  ( n  e.  NN0  /\  F : ( 1 ... n ) -1-1-onto-> A ) )  ->  ( 0 ... ( # `  F
) )  =  ( 0 ... n ) )
3029feq2d 5718 . . . . . . . . 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 223 . . . . . . . 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 1177 . . . . . . 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 3064 . . . . . . . 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 223 . . . . . . 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 563 . . . . . 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 436 . . . . 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 615 . . . 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 1210 . . 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 2955 . 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   {cpr 4029   class class class wbr 4447   dom cdm 4999    Fn wfn 5583   -->wf 5584   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284    ~~ cen 7513   Fincfn 7516   0cc0 9492   1c1 9493    - cmin 9805   NN0cn0 10795   ...cfz 11672   #chash 12373   UMGrph cumg 24016   EulPaths ceup 24666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-er 7311  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-hash 12374  df-umgra 24017  df-eupa 24667
This theorem is referenced by:  eupacl  24673  eupaf1o  24674  eupapf  24676  eupaseg  24677
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