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Theorem iswlkon 25274
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 25273 . . . 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 4416 . . 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 1028 . 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 4410 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( f ( V Walks 
E ) p  <->  F ( V Walks  E ) P ) )
5 fveq1 5869 . . . . . . 7  |-  ( p  =  P  ->  (
p `  0 )  =  ( P ` 
0 ) )
65eqeq1d 2455 . . . . . 6  |-  ( p  =  P  ->  (
( p `  0
)  =  A  <->  ( P `  0 )  =  A ) )
76adantl 468 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( p ` 
0 )  =  A  <-> 
( P `  0
)  =  A ) )
8 simpr 463 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  p  =  P )
9 fveq2 5870 . . . . . . . 8  |-  ( f  =  F  ->  ( # `
 f )  =  ( # `  F
) )
109adantr 467 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  ( # `  f
)  =  ( # `  F ) )
118, 10fveq12d 5876 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( p `  ( # `
 f ) )  =  ( P `  ( # `  F ) ) )
1211eqeq1d 2455 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( p `  ( # `  f ) )  =  B  <->  ( P `  ( # `  F
) )  =  B ) )
134, 7, 123anbi123d 1341 . . . 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 2453 . . . 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 4718 . . 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 1031 . 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 257 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 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   class class class wbr 4405   {copab 4463   ` cfv 5585  (class class class)co 6295   0cc0 9544   #chash 12522   Walks cwalk 25238   WalkOn cwlkon 25242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  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-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12523  df-word 12671  df-wlk 25248  df-wlkon 25254
This theorem is referenced by:  wlkonprop  25275  wlkonwlk  25277  0wlkon  25289  isspthonpth  25326  spthonepeq  25329  1pthon  25333  2pthon  25344  usgra2adedgwlkon  25355  usgra2adedgwlkonALT  25356  el2wlkonot  25609  el2spthonot  25610
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