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Theorem isspthonpth 21584
Description: Properties of a pair of functions to be a simple path between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 9-Mar-2018.)
Assertion
Ref Expression
isspthonpth  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z )  /\  ( A  e.  V  /\  B  e.  V )
)  ->  ( F
( A ( V SPathOn  E ) B ) P  <->  ( F ( V SPaths  E ) P  /\  ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) ) )

Proof of Theorem isspthonpth
StepHypRef Expression
1 isspthon 21583 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z )  /\  ( A  e.  V  /\  B  e.  V )
)  ->  ( F
( A ( V SPathOn  E ) B ) P  <->  ( F ( A ( V WalkOn  E
) B ) P  /\  F ( V SPaths  E ) P ) ) )
2 iswlkon 21531 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z )  /\  ( A  e.  V  /\  B  e.  V )
)  ->  ( F
( A ( V WalkOn  E ) B ) P  <->  ( F ( V Walks  E ) P  /\  ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) ) )
32anbi1d 686 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z )  /\  ( A  e.  V  /\  B  e.  V )
)  ->  ( ( F ( A ( V WalkOn  E ) B ) P  /\  F
( V SPaths  E ) P )  <->  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  /\  F ( V SPaths  E ) P ) ) )
4 simpl 444 . . . . . 6  |-  ( ( F ( V SPaths  E
) P  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  ->  F
( V SPaths  E ) P )
5 simpr2 964 . . . . . 6  |-  ( ( F ( V SPaths  E
) P  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  ->  ( P `  0 )  =  A )
6 simpr3 965 . . . . . 6  |-  ( ( F ( V SPaths  E
) P  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  ->  ( P `  ( # `  F
) )  =  B )
74, 5, 63jca 1134 . . . . 5  |-  ( ( F ( V SPaths  E
) P  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  ->  ( F ( V SPaths  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )
87ancoms 440 . . . 4  |-  ( ( ( F ( V Walks 
E ) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  /\  F ( V SPaths  E ) P )  ->  ( F
( V SPaths  E ) P  /\  ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )
9 spthispth 21573 . . . . . . 7  |-  ( F ( V SPaths  E ) P  ->  F ( V Paths  E ) P )
10 pthistrl 21572 . . . . . . 7  |-  ( F ( V Paths  E ) P  ->  F ( V Trails  E ) P )
11 trliswlk 21539 . . . . . . 7  |-  ( F ( V Trails  E ) P  ->  F ( V Walks  E ) P )
129, 10, 113syl 19 . . . . . 6  |-  ( F ( V SPaths  E ) P  ->  F ( V Walks  E ) P )
13123anim1i 1140 . . . . 5  |-  ( ( F ( V SPaths  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  ( F
( V Walks  E ) P  /\  ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )
14 simp1 957 . . . . 5  |-  ( ( F ( V SPaths  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  F ( V SPaths  E ) P )
1513, 14jca 519 . . . 4  |-  ( ( F ( V SPaths  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  /\  F ( V SPaths  E ) P ) )
168, 15impbii 181 . . 3  |-  ( ( ( F ( V Walks 
E ) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  /\  F ( V SPaths  E ) P )  <->  ( F ( V SPaths  E ) P  /\  ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )
173, 16syl6bb 253 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z )  /\  ( A  e.  V  /\  B  e.  V )
)  ->  ( ( F ( A ( V WalkOn  E ) B ) P  /\  F
( V SPaths  E ) P )  <->  ( F
( V SPaths  E ) P  /\  ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) ) )
181, 17bitrd 245 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z )  /\  ( A  e.  V  /\  B  e.  V )
)  ->  ( F
( A ( V SPathOn  E ) B ) P  <->  ( F ( V SPaths  E ) P  /\  ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   0cc0 8990   #chash 11618   Walks cwalk 21506   Trails ctrail 21507   Paths cpath 21508   SPaths cspath 21509   WalkOn cwlkon 21510   SPathOn cspthon 21513
This theorem is referenced by:  el2spthonot0  28338
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-hash 11619  df-word 11723  df-wlk 21516  df-trail 21517  df-pth 21518  df-spth 21519  df-wlkon 21522  df-spthon 21525
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