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Theorem iswlkon 23442
Description: Properties of a pair of functions to be a walk between two given vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 2-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
Assertion
Ref Expression
iswlkon  |-  ( ( ( 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 ) ) )

Proof of Theorem iswlkon
Dummy variables  f  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wlkon 23441 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V WalkOn  E ) B )  =  { <. f ,  p >.  |  (
f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  B ) } )
21breqd 4315 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( F ( A ( V WalkOn  E ) B ) P  <->  F { <. f ,  p >.  |  ( f ( V Walks 
E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  B ) } P ) )
323adant2 1007 . 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 { <. f ,  p >.  |  (
f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  B ) } P ) )
4 breq12 4309 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( f ( V Walks 
E ) p  <->  F ( V Walks  E ) P ) )
5 fveq1 5702 . . . . . . 7  |-  ( p  =  P  ->  (
p `  0 )  =  ( P ` 
0 ) )
65eqeq1d 2451 . . . . . 6  |-  ( p  =  P  ->  (
( p `  0
)  =  A  <->  ( P `  0 )  =  A ) )
76adantl 466 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( p ` 
0 )  =  A  <-> 
( P `  0
)  =  A ) )
8 simpr 461 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  p  =  P )
9 fveq2 5703 . . . . . . . 8  |-  ( f  =  F  ->  ( # `
 f )  =  ( # `  F
) )
109adantr 465 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  ( # `  f
)  =  ( # `  F ) )
118, 10fveq12d 5709 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( p `  ( # `
 f ) )  =  ( P `  ( # `  F ) ) )
1211eqeq1d 2451 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( p `  ( # `  f ) )  =  B  <->  ( P `  ( # `  F
) )  =  B ) )
134, 7, 123anbi123d 1289 . . . 4  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  B )  <-> 
( F ( V Walks 
E ) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) ) )
14 eqid 2443 . . . 4  |-  { <. f ,  p >.  |  ( f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  B ) }  =  { <. f ,  p >.  |  ( f ( V Walks 
E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  B ) }
1513, 14brabga 4615 . . 3  |-  ( ( F  e.  W  /\  P  e.  Z )  ->  ( F { <. f ,  p >.  |  ( f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  B ) } P  <->  ( F
( V Walks  E ) P  /\  ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) ) )
16153ad2ant2 1010 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( F  e.  W  /\  P  e.  Z )  /\  ( A  e.  V  /\  B  e.  V )
)  ->  ( F { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  B ) } P  <->  ( F
( V Walks  E ) P  /\  ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) ) )
173, 16bitrd 253 1  |-  ( ( ( 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 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4304   {copab 4361   ` cfv 5430  (class class class)co 6103   0cc0 9294   #chash 12115   Walks cwalk 23417   WalkOn cwlkon 23421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450  df-fzo 11561  df-word 12241  df-wlk 23427  df-wlkon 23433
This theorem is referenced by:  wlkonprop  23443  wlkonwlk  23446  0wlkon  23458  isspthonpth  23495  spthonepeq  23498  1pthon  23502  2pthon  23513  usgra2adedgwlkon  30319  el2wlkonot  30400  el2spthonot  30401
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