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Theorem isspthonpth 21537
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 21536 . 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 21484 . . . 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 21526 . . . . . . 7  |-  ( F ( V SPaths  E ) P  ->  F ( V Paths  E ) P )
10 pthistrl 21525 . . . . . . 7  |-  ( F ( V Paths  E ) P  ->  F ( V Trails  E ) P )
11 trliswlk 21492 . . . . . . 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 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   0cc0 8946   #chash 11573   Walks cwalk 21459   Trails ctrail 21460   Paths cpath 21461   SPaths cspath 21462   WalkOn cwlkon 21463   SPathOn cspthon 21466
This theorem is referenced by:  el2spthonot0  28068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-hash 11574  df-word 11678  df-wlk 21469  df-trail 21470  df-pth 21471  df-spth 21472  df-wlkon 21475  df-spthon 21478
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