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Theorem pthonprop 24700
Description: Properties of a path between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
Assertion
Ref Expression
pthonprop  |-  ( F ( A ( V PathOn  E ) B ) P  ->  ( (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( A ( V WalkOn  E ) B ) P  /\  F
( V Paths  E ) P ) ) )

Proof of Theorem pthonprop
Dummy variables  a 
b  e  f  p  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pthon 24637 . . 3  |- PathOn  =  ( v  e.  _V , 
e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  {
<. f ,  p >.  |  ( f ( a ( v WalkOn  e ) b ) p  /\  f ( v Paths  e
) p ) } ) )
2 oveq12 6205 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v WalkOn  e )  =  ( V WalkOn  E
) )
32oveqd 6213 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( a ( v WalkOn 
e ) b )  =  ( a ( V WalkOn  E ) b ) )
43breqd 4378 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( a ( v WalkOn  e ) b ) p  <->  f (
a ( V WalkOn  E
) b ) p ) )
5 oveq12 6205 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v Paths  e )  =  ( V Paths  E
) )
65breqd 4378 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( v Paths 
e ) p  <->  f ( V Paths  E ) p ) )
74, 6anbi12d 708 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( f ( a ( v WalkOn  e
) b ) p  /\  f ( v Paths 
e ) p )  <-> 
( f ( a ( V WalkOn  E ) b ) p  /\  f ( V Paths  E
) p ) ) )
8 pthon 24698 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( A ( V PathOn  E
) B )  =  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E
) B ) p  /\  f ( V Paths 
E ) p ) } )
91, 7, 8bropopvvv 6779 . 2  |-  ( F ( A ( V PathOn  E ) B ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )
) )
10 ispthon 24699 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( F ( A ( V PathOn  E ) B ) P  <->  ( F
( A ( V WalkOn  E ) B ) P  /\  F ( V Paths  E ) P ) ) )
1110biimpd 207 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( F ( A ( V PathOn  E ) B ) P  ->  ( F ( A ( V WalkOn  E ) B ) P  /\  F
( V Paths  E ) P ) ) )
1211imdistani 688 . 2  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )
)  /\  F ( A ( V PathOn  E
) B ) P )  ->  ( (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( A ( V WalkOn  E ) B ) P  /\  F
( V Paths  E ) P ) ) )
139, 12mpancom 667 1  |-  ( F ( A ( V PathOn  E ) B ) P  ->  ( (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( A ( V WalkOn  E ) B ) P  /\  F
( V Paths  E ) P ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   _Vcvv 3034   class class class wbr 4367  (class class class)co 6196   Paths cpath 24621   WalkOn cwlkon 24623   PathOn cpthon 24625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-hash 12308  df-word 12446  df-wlk 24629  df-trail 24630  df-pth 24631  df-pthon 24637
This theorem is referenced by:  pthonispth  24701
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