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Theorem ispth 24246
Description: Properties of a pair of functions to be a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
Assertion
Ref Expression
ispth  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( F ( V Paths 
E ) P  <->  ( F
( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) ) )

Proof of Theorem ispth
Dummy variables  f  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4448 . . 3  |-  ( F ( V Paths  E ) P  <->  <. F ,  P >.  e.  ( V Paths  E
) )
2 pths 24244 . . . . 5  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V Paths  E )  =  { <. f ,  p >.  |  (
f ( V Trails  E
) p  /\  Fun  `' ( p  |`  (
1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) } )
32adantr 465 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( V Paths  E )  =  { <. f ,  p >.  |  ( f ( V Trails  E ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) } )
43eleq2d 2537 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( <. F ,  P >.  e.  ( V Paths  E
)  <->  <. F ,  P >.  e.  { <. f ,  p >.  |  (
f ( V Trails  E
) p  /\  Fun  `' ( p  |`  (
1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) } ) )
51, 4syl5bb 257 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( F ( V Paths 
E ) P  <->  <. F ,  P >.  e.  { <. f ,  p >.  |  ( f ( V Trails  E
) p  /\  Fun  `' ( p  |`  (
1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) } ) )
6 breq12 4452 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( f ( V Trails  E ) p  <->  F ( V Trails  E ) P ) )
7 simpr 461 . . . . . . . 8  |-  ( ( f  =  F  /\  p  =  P )  ->  p  =  P )
8 fveq2 5864 . . . . . . . . . 10  |-  ( f  =  F  ->  ( # `
 f )  =  ( # `  F
) )
98adantr 465 . . . . . . . . 9  |-  ( ( f  =  F  /\  p  =  P )  ->  ( # `  f
)  =  ( # `  F ) )
109oveq2d 6298 . . . . . . . 8  |-  ( ( f  =  F  /\  p  =  P )  ->  ( 1..^ ( # `  f ) )  =  ( 1..^ ( # `  F ) ) )
117, 10reseq12d 5272 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  ( p  |`  (
1..^ ( # `  f
) ) )  =  ( P  |`  (
1..^ ( # `  F
) ) ) )
1211cnveqd 5176 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  `' ( p  |`  ( 1..^ ( # `  f
) ) )  =  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )
1312funeqd 5607 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( Fun  `' ( p  |`  ( 1..^ ( # `  f
) ) )  <->  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) ) )
148preq2d 4113 . . . . . . . . 9  |-  ( f  =  F  ->  { 0 ,  ( # `  f
) }  =  {
0 ,  ( # `  F ) } )
1514adantr 465 . . . . . . . 8  |-  ( ( f  =  F  /\  p  =  P )  ->  { 0 ,  (
# `  f ) }  =  { 0 ,  ( # `  F
) } )
167, 15imaeq12d 5336 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  ( p " {
0 ,  ( # `  f ) } )  =  ( P " { 0 ,  (
# `  F ) } ) )
178oveq2d 6298 . . . . . . . . 9  |-  ( f  =  F  ->  (
1..^ ( # `  f
) )  =  ( 1..^ ( # `  F
) ) )
1817adantr 465 . . . . . . . 8  |-  ( ( f  =  F  /\  p  =  P )  ->  ( 1..^ ( # `  f ) )  =  ( 1..^ ( # `  F ) ) )
197, 18imaeq12d 5336 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  ( p " (
1..^ ( # `  f
) ) )  =  ( P " (
1..^ ( # `  F
) ) ) )
2016, 19ineq12d 3701 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) ) )
2120eqeq1d 2469 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( ( p
" { 0 ,  ( # `  f
) } )  i^i  ( p " (
1..^ ( # `  f
) ) ) )  =  (/)  <->  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) ) )
226, 13, 213anbi123d 1299 . . . 4  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( f ( V Trails  E ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) )  <->  ( F
( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) ) )
2322opelopabga 4760 . . 3  |-  ( ( F  e.  W  /\  P  e.  Z )  ->  ( <. F ,  P >.  e.  { <. f ,  p >.  |  (
f ( V Trails  E
) p  /\  Fun  `' ( p  |`  (
1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) }  <->  ( F
( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) ) )
2423adantl 466 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( <. F ,  P >.  e.  { <. f ,  p >.  |  (
f ( V Trails  E
) p  /\  Fun  `' ( p  |`  (
1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) }  <->  ( F
( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) ) )
255, 24bitrd 253 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z ) )  -> 
( F ( V Paths 
E ) P  <->  ( F
( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    i^i cin 3475   (/)c0 3785   {cpr 4029   <.cop 4033   class class class wbr 4447   {copab 4504   `'ccnv 4998    |` cres 5001   "cima 5002   Fun wfun 5580   ` cfv 5586  (class class class)co 6282   0cc0 9488   1c1 9489  ..^cfzo 11788   #chash 12369   Trails ctrail 24175   Paths cpath 24176
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 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-word 12504  df-wlk 24184  df-trail 24185  df-pth 24186
This theorem is referenced by:  0pth  24248  pthistrl  24250  spthispth  24251  pthdepisspth  24252  1pthon  24269  constr2pth  24279  3v3e3cycl1  24320  constr3pth  24336  4cycl4v4e  24342  4cycl4dv4e  24344  usgra2pthspth  31820
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