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Theorem wlknwwlknvbij 24402
 Description: There is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length and starting at the same vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
Assertion
Ref Expression
wlknwwlknvbij USGrph Walks WWalksN
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,

Proof of Theorem wlknwwlknvbij
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6300 . . . . . 6 Walks
21rabex 4591 . . . . 5 Walks
32mptex 6122 . . . 4 Walks
43resex 5308 . . 3 Walks Walks
5 eqid 2460 . . . 4 Walks Walks
6 fveq2 5857 . . . . . . . . 9
76fveq2d 5861 . . . . . . . 8
87eqeq1d 2462 . . . . . . 7
98cbvrabv 3105 . . . . . 6 Walks Walks
10 eqid 2460 . . . . . 6 WWalksN WWalksN
11 fveq2 5857 . . . . . . 7
1211cbvmptv 4531 . . . . . 6 Walks Walks
139, 10, 12wlknwwlknbij 24375 . . . . 5 USGrph Walks Walks WWalksN
14133adant3 1011 . . . 4 USGrph Walks Walks WWalksN
15 fveq1 5856 . . . . . 6
1615eqeq1d 2462 . . . . 5
17163ad2ant3 1014 . . . 4 USGrph Walks
185, 14, 17f1oresrab 6044 . . 3 USGrph Walks Walks Walks WWalksN
19 f1oeq1 5798 . . . 4 Walks Walks Walks WWalksN Walks Walks Walks WWalksN
2019spcegv 3192 . . 3 Walks Walks Walks Walks Walks WWalksN Walks WWalksN
214, 18, 20mpsyl 63 . 2 USGrph Walks WWalksN
22 df-rab 2816 . . . . 5 Walks Walks
23 anass 649 . . . . . . 7 Walks Walks
2423bicomi 202 . . . . . 6 Walks Walks
2524abbii 2594 . . . . 5 Walks Walks
26 fveq2 5857 . . . . . . . . . . . 12
2726fveq2d 5861 . . . . . . . . . . 11
2827eqeq1d 2462 . . . . . . . . . 10
2928elrab 3254 . . . . . . . . 9 Walks Walks
3029anbi1i 695 . . . . . . . 8 Walks Walks
3130bicomi 202 . . . . . . 7 Walks Walks
3231abbii 2594 . . . . . 6 Walks Walks
33 df-rab 2816 . . . . . 6 Walks Walks
3432, 33eqtr4i 2492 . . . . 5 Walks Walks
3522, 25, 343eqtri 2493 . . . 4 Walks Walks
36 f1oeq2 5799 . . . 4 Walks Walks Walks WWalksN Walks WWalksN
3735, 36mp1i 12 . . 3 USGrph Walks WWalksN Walks WWalksN
3837exbidv 1685 . 2 USGrph Walks WWalksN Walks WWalksN
3921, 38mpbird 232 1 USGrph Walks WWalksN
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 968   wceq 1374  wex 1591   wcel 1762  cab 2445  crab 2811  cvv 3106   class class class wbr 4440   cmpt 4498   cres 4994  wf1o 5578  cfv 5579  (class class class)co 6275  c1st 6772  c2nd 6773  cc0 9481  cn0 10784  chash 12360   USGrph cusg 23993   Walks cwalk 24160   WWalksN cwwlkn 24340 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-usgra 23996  df-wlk 24170  df-wwlk 24341  df-wwlkn 24342 This theorem is referenced by:  rusgranumwwlkg  24621
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