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Theorem spths 24690
Description: The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
Assertion
Ref Expression
spths  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V SPaths  E )  =  { <. f ,  p >.  |  (
f ( V Trails  E
) p  /\  Fun  `' p ) } )
Distinct variable groups:    f, E, p    f, V, p
Allowed substitution hints:    X( f, p)    Y( f, p)

Proof of Theorem spths
Dummy variables  e 
v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3043 . 2  |-  ( V  e.  X  ->  V  e.  _V )
2 elex 3043 . 2  |-  ( E  e.  Y  ->  E  e.  _V )
3 df-spth 24632 . . . 4  |- SPaths  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Trails 
e ) p  /\  Fun  `' p ) } )
43a1i 11 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  -> SPaths 
=  ( v  e. 
_V ,  e  e. 
_V  |->  { <. f ,  p >.  |  (
f ( v Trails  e
) p  /\  Fun  `' p ) } ) )
5 oveq12 6205 . . . . . . 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 4378 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
f ( v Trails  e
) p  <->  f ( V Trails  E ) p ) )
87anbi1d 702 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
( f ( v Trails 
e ) p  /\  Fun  `' p )  <->  ( f
( V Trails  E )
p  /\  Fun  `' p
) ) )
98opabbidv 4430 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  { <. f ,  p >.  |  ( f ( v Trails  e
) p  /\  Fun  `' p ) }  =  { <. f ,  p >.  |  ( f ( V Trails  E ) p  /\  Fun  `' p
) } )
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 trliswlk 24662 . . . 4  |-  ( f ( V Trails  E ) p  ->  f ( V Walks  E ) p )
1312wlkres 24643 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  ( f ( V Trails  E ) p  /\  Fun  `' p
) }  e.  _V )
144, 9, 10, 11, 13ovmpt2d 6329 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V SPaths  E )  =  { <. f ,  p >.  |  (
f ( V Trails  E
) p  /\  Fun  `' p ) } )
151, 2, 14syl2an 475 1  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V SPaths  E )  =  { <. f ,  p >.  |  (
f ( V Trails  E
) p  /\  Fun  `' p ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   _Vcvv 3034   class class class wbr 4367   {copab 4424   `'ccnv 4912   Fun wfun 5490  (class class class)co 6196    |-> cmpt2 6198   Trails ctrail 24620   SPaths cspath 24622
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-spth 24632
This theorem is referenced by:  isspth  24692
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