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Theorem pths 24770
Description: The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Assertion
Ref Expression
pths  |-  ( ( 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
) ) ) )  =  (/) ) } )
Distinct variable groups:    f, E, p    f, V, p
Allowed substitution hints:    X( f, p)    Y( f, p)

Proof of Theorem pths
Dummy variables  e 
v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3115 . 2  |-  ( V  e.  X  ->  V  e.  _V )
2 elex 3115 . 2  |-  ( E  e.  Y  ->  E  e.  _V )
3 df-pth 24712 . . . 4  |- Paths  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Trails 
e ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) } )
43a1i 11 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  -> Paths  =  ( v  e. 
_V ,  e  e. 
_V  |->  { <. f ,  p >.  |  (
f ( v Trails  e
) p  /\  Fun  `' ( p  |`  (
1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) } ) )
5 oveq12 6279 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Trails  e )  =  ( V Trails  E
) )
65adantl 464 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
v Trails  e )  =  ( V Trails  E ) )
76breqd 4450 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
f ( v Trails  e
) p  <->  f ( V Trails  E ) p ) )
873anbi1d 1301 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
( 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
) ) ) )  =  (/) ) ) )
98opabbidv 4502 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  { <. f ,  p >.  |  ( f ( v Trails  e
) p  /\  Fun  `' ( p  |`  (
1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) }  =  { <. f ,  p >.  |  ( f ( V Trails  E ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) } )
10 simpl 455 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  V  e.  _V )
11 simpr 459 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  E  e.  _V )
12 3anass 975 . . . . . 6  |-  ( ( 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
) ) ) )  =  (/) ) ) )
1312a1i 11 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( ( 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
) ) ) )  =  (/) ) ) ) )
1413opabbidv 4502 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  ( f ( V Trails  E ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) }  =  { <. f ,  p >.  |  ( f ( V Trails  E ) p  /\  ( Fun  `' ( p  |`  ( 1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) ) } )
15 trliswlk 24743 . . . . 5  |-  ( f ( V Trails  E ) p  ->  f ( V Walks  E ) p )
1615wlkres 24724 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  ( f ( V Trails  E ) p  /\  ( Fun  `' ( p  |`  ( 1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) ) }  e.  _V )
1714, 16eqeltrd 2542 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  ( f ( V Trails  E ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) }  e.  _V )
184, 9, 10, 11, 17ovmpt2d 6403 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V Paths  E )  =  { <. f ,  p >.  |  (
f ( V Trails  E
) p  /\  Fun  `' ( p  |`  (
1..^ ( # `  f
) ) )  /\  ( ( p " { 0 ,  (
# `  f ) } )  i^i  (
p " ( 1..^ ( # `  f
) ) ) )  =  (/) ) } )
191, 2, 18syl2an 475 1  |-  ( ( 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
) ) ) )  =  (/) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106    i^i cin 3460   (/)c0 3783   {cpr 4018   class class class wbr 4439   {copab 4496   `'ccnv 4987    |` cres 4990   "cima 4991   Fun wfun 5564   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   0cc0 9481   1c1 9482  ..^cfzo 11799   #chash 12387   Trails ctrail 24701   Paths cpath 24702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-wlk 24710  df-trail 24711  df-pth 24712
This theorem is referenced by:  ispth  24772
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