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Theorem ispth 24974
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 4395 . . 3  |-  ( F ( V Paths  E ) P  <->  <. F ,  P >.  e.  ( V Paths  E
) )
2 pths 24972 . . . . 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 463 . . . 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 2472 . . 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 4399 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( f ( V Trails  E ) p  <->  F ( V Trails  E ) P ) )
7 simpr 459 . . . . . . . 8  |-  ( ( f  =  F  /\  p  =  P )  ->  p  =  P )
8 fveq2 5848 . . . . . . . . . 10  |-  ( f  =  F  ->  ( # `
 f )  =  ( # `  F
) )
98adantr 463 . . . . . . . . 9  |-  ( ( f  =  F  /\  p  =  P )  ->  ( # `  f
)  =  ( # `  F ) )
109oveq2d 6293 . . . . . . . 8  |-  ( ( f  =  F  /\  p  =  P )  ->  ( 1..^ ( # `  f ) )  =  ( 1..^ ( # `  F ) ) )
117, 10reseq12d 5094 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  ( p  |`  (
1..^ ( # `  f
) ) )  =  ( P  |`  (
1..^ ( # `  F
) ) ) )
1211cnveqd 4998 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  `' ( p  |`  ( 1..^ ( # `  f
) ) )  =  `' ( P  |`  ( 1..^ ( # `  F
) ) ) )
1312funeqd 5589 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( Fun  `' ( p  |`  ( 1..^ ( # `  f
) ) )  <->  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) ) ) )
148preq2d 4057 . . . . . . . . 9  |-  ( f  =  F  ->  { 0 ,  ( # `  f
) }  =  {
0 ,  ( # `  F ) } )
1514adantr 463 . . . . . . . 8  |-  ( ( f  =  F  /\  p  =  P )  ->  { 0 ,  (
# `  f ) }  =  { 0 ,  ( # `  F
) } )
167, 15imaeq12d 5157 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  ( p " {
0 ,  ( # `  f ) } )  =  ( P " { 0 ,  (
# `  F ) } ) )
178oveq2d 6293 . . . . . . . . 9  |-  ( f  =  F  ->  (
1..^ ( # `  f
) )  =  ( 1..^ ( # `  F
) ) )
1817adantr 463 . . . . . . . 8  |-  ( ( f  =  F  /\  p  =  P )  ->  ( 1..^ ( # `  f ) )  =  ( 1..^ ( # `  F ) ) )
197, 18imaeq12d 5157 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  ( p " (
1..^ ( # `  f
) ) )  =  ( P " (
1..^ ( # `  F
) ) ) )
2016, 19ineq12d 3641 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) ) )
2120eqeq1d 2404 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( ( p
" { 0 ,  ( # `  f
) } )  i^i  ( p " (
1..^ ( # `  f
) ) ) )  =  (/)  <->  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) ) )
226, 13, 213anbi123d 1301 . . . 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 4702 . . 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 464 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    i^i cin 3412   (/)c0 3737   {cpr 3973   <.cop 3977   class class class wbr 4394   {copab 4451   `'ccnv 4821    |` cres 4824   "cima 4825   Fun wfun 5562   ` cfv 5568  (class class class)co 6277   0cc0 9521   1c1 9522  ..^cfzo 11852   #chash 12450   Trails ctrail 24903   Paths cpath 24904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-fzo 11853  df-hash 12451  df-word 12589  df-wlk 24912  df-trail 24913  df-pth 24914
This theorem is referenced by:  0pth  24976  pthistrl  24978  spthispth  24979  pthdepisspth  24980  1pthon  24997  constr2pth  25007  3v3e3cycl1  25048  constr3pth  25064  4cycl4v4e  25070  4cycl4dv4e  25072  usgra2pthspth  37961
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