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Theorem wlknwwlksnbij 29406
Description: The mapping (𝑑 ∈ 𝑇 ↦ (2nd β€˜π‘‘)) is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length in a simple pseudograph. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 5-Aug-2022.)
Hypotheses
Ref Expression
wlknwwlksnbij.t 𝑇 = {𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}
wlknwwlksnbij.w π‘Š = (𝑁 WWalksN 𝐺)
wlknwwlksnbij.f 𝐹 = (𝑑 ∈ 𝑇 ↦ (2nd β€˜π‘‘))
Assertion
Ref Expression
wlknwwlksnbij ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ 𝐹:𝑇–1-1-ontoβ†’π‘Š)
Distinct variable groups:   𝐺,𝑝,𝑑   𝑁,𝑝,𝑑   𝑑,𝑇
Allowed substitution hints:   𝑇(𝑝)   𝐹(𝑑,𝑝)   π‘Š(𝑑,𝑝)

Proof of Theorem wlknwwlksnbij
Dummy variable π‘ž is distinct from all other variables.
StepHypRef Expression
1 eqid 2731 . . 3 (𝑝 ∈ (Walksβ€˜πΊ) ↦ (2nd β€˜π‘)) = (𝑝 ∈ (Walksβ€˜πΊ) ↦ (2nd β€˜π‘))
21wlkswwlksf1o 29397 . . . 4 (𝐺 ∈ USPGraph β†’ (𝑝 ∈ (Walksβ€˜πΊ) ↦ (2nd β€˜π‘)):(Walksβ€˜πΊ)–1-1-ontoβ†’(WWalksβ€˜πΊ))
32adantr 480 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ (𝑝 ∈ (Walksβ€˜πΊ) ↦ (2nd β€˜π‘)):(Walksβ€˜πΊ)–1-1-ontoβ†’(WWalksβ€˜πΊ))
4 fveqeq2 6901 . . . . 5 (π‘ž = (2nd β€˜π‘) β†’ ((β™―β€˜π‘ž) = (𝑁 + 1) ↔ (β™―β€˜(2nd β€˜π‘)) = (𝑁 + 1)))
543ad2ant3 1134 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) ∧ 𝑝 ∈ (Walksβ€˜πΊ) ∧ π‘ž = (2nd β€˜π‘)) β†’ ((β™―β€˜π‘ž) = (𝑁 + 1) ↔ (β™―β€˜(2nd β€˜π‘)) = (𝑁 + 1)))
6 wlkcpr 29150 . . . . . . 7 (𝑝 ∈ (Walksβ€˜πΊ) ↔ (1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘))
7 wlklenvp1 29139 . . . . . . . 8 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (β™―β€˜(2nd β€˜π‘)) = ((β™―β€˜(1st β€˜π‘)) + 1))
8 eqeq1 2735 . . . . . . . . . 10 ((β™―β€˜(2nd β€˜π‘)) = ((β™―β€˜(1st β€˜π‘)) + 1) β†’ ((β™―β€˜(2nd β€˜π‘)) = (𝑁 + 1) ↔ ((β™―β€˜(1st β€˜π‘)) + 1) = (𝑁 + 1)))
9 wlkcl 29136 . . . . . . . . . . . . 13 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (β™―β€˜(1st β€˜π‘)) ∈ β„•0)
109nn0cnd 12539 . . . . . . . . . . . 12 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (β™―β€˜(1st β€˜π‘)) ∈ β„‚)
1110adantr 480 . . . . . . . . . . 11 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0)) β†’ (β™―β€˜(1st β€˜π‘)) ∈ β„‚)
12 nn0cn 12487 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„‚)
1312adantl 481 . . . . . . . . . . . 12 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ 𝑁 ∈ β„‚)
1413adantl 481 . . . . . . . . . . 11 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0)) β†’ 𝑁 ∈ β„‚)
15 1cnd 11214 . . . . . . . . . . 11 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0)) β†’ 1 ∈ β„‚)
1611, 14, 15addcan2d 11423 . . . . . . . . . 10 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0)) β†’ (((β™―β€˜(1st β€˜π‘)) + 1) = (𝑁 + 1) ↔ (β™―β€˜(1st β€˜π‘)) = 𝑁))
178, 16sylan9bbr 510 . . . . . . . . 9 ((((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜(2nd β€˜π‘)) = ((β™―β€˜(1st β€˜π‘)) + 1)) β†’ ((β™―β€˜(2nd β€˜π‘)) = (𝑁 + 1) ↔ (β™―β€˜(1st β€˜π‘)) = 𝑁))
1817exp31 419 . . . . . . . 8 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ ((β™―β€˜(2nd β€˜π‘)) = ((β™―β€˜(1st β€˜π‘)) + 1) β†’ ((β™―β€˜(2nd β€˜π‘)) = (𝑁 + 1) ↔ (β™―β€˜(1st β€˜π‘)) = 𝑁))))
197, 18mpid 44 . . . . . . 7 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ ((β™―β€˜(2nd β€˜π‘)) = (𝑁 + 1) ↔ (β™―β€˜(1st β€˜π‘)) = 𝑁)))
206, 19sylbi 216 . . . . . 6 (𝑝 ∈ (Walksβ€˜πΊ) β†’ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ ((β™―β€˜(2nd β€˜π‘)) = (𝑁 + 1) ↔ (β™―β€˜(1st β€˜π‘)) = 𝑁)))
2120impcom 407 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) ∧ 𝑝 ∈ (Walksβ€˜πΊ)) β†’ ((β™―β€˜(2nd β€˜π‘)) = (𝑁 + 1) ↔ (β™―β€˜(1st β€˜π‘)) = 𝑁))
22213adant3 1131 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) ∧ 𝑝 ∈ (Walksβ€˜πΊ) ∧ π‘ž = (2nd β€˜π‘)) β†’ ((β™―β€˜(2nd β€˜π‘)) = (𝑁 + 1) ↔ (β™―β€˜(1st β€˜π‘)) = 𝑁))
235, 22bitrd 278 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) ∧ 𝑝 ∈ (Walksβ€˜πΊ) ∧ π‘ž = (2nd β€˜π‘)) β†’ ((β™―β€˜π‘ž) = (𝑁 + 1) ↔ (β™―β€˜(1st β€˜π‘)) = 𝑁))
241, 3, 23f1oresrab 7128 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ ((𝑝 ∈ (Walksβ€˜πΊ) ↦ (2nd β€˜π‘)) β†Ύ {𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}):{𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}–1-1-ontoβ†’{π‘ž ∈ (WWalksβ€˜πΊ) ∣ (β™―β€˜π‘ž) = (𝑁 + 1)})
25 wlknwwlksnbij.f . . . 4 𝐹 = (𝑑 ∈ 𝑇 ↦ (2nd β€˜π‘‘))
26 wlknwwlksnbij.t . . . . . . 7 𝑇 = {𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}
2726mpteq1i 5245 . . . . . 6 (𝑑 ∈ 𝑇 ↦ (2nd β€˜π‘‘)) = (𝑑 ∈ {𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁} ↦ (2nd β€˜π‘‘))
28 ssrab2 4078 . . . . . . 7 {𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁} βŠ† (Walksβ€˜πΊ)
29 resmpt 6038 . . . . . . 7 ({𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁} βŠ† (Walksβ€˜πΊ) β†’ ((𝑑 ∈ (Walksβ€˜πΊ) ↦ (2nd β€˜π‘‘)) β†Ύ {𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}) = (𝑑 ∈ {𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁} ↦ (2nd β€˜π‘‘)))
3028, 29ax-mp 5 . . . . . 6 ((𝑑 ∈ (Walksβ€˜πΊ) ↦ (2nd β€˜π‘‘)) β†Ύ {𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}) = (𝑑 ∈ {𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁} ↦ (2nd β€˜π‘‘))
31 fveq2 6892 . . . . . . . 8 (𝑑 = 𝑝 β†’ (2nd β€˜π‘‘) = (2nd β€˜π‘))
3231cbvmptv 5262 . . . . . . 7 (𝑑 ∈ (Walksβ€˜πΊ) ↦ (2nd β€˜π‘‘)) = (𝑝 ∈ (Walksβ€˜πΊ) ↦ (2nd β€˜π‘))
3332reseq1i 5978 . . . . . 6 ((𝑑 ∈ (Walksβ€˜πΊ) ↦ (2nd β€˜π‘‘)) β†Ύ {𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}) = ((𝑝 ∈ (Walksβ€˜πΊ) ↦ (2nd β€˜π‘)) β†Ύ {𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁})
3427, 30, 333eqtr2i 2765 . . . . 5 (𝑑 ∈ 𝑇 ↦ (2nd β€˜π‘‘)) = ((𝑝 ∈ (Walksβ€˜πΊ) ↦ (2nd β€˜π‘)) β†Ύ {𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁})
3534a1i 11 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ (𝑑 ∈ 𝑇 ↦ (2nd β€˜π‘‘)) = ((𝑝 ∈ (Walksβ€˜πΊ) ↦ (2nd β€˜π‘)) β†Ύ {𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}))
3625, 35eqtrid 2783 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ 𝐹 = ((𝑝 ∈ (Walksβ€˜πΊ) ↦ (2nd β€˜π‘)) β†Ύ {𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}))
3726a1i 11 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ 𝑇 = {𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁})
38 wlknwwlksnbij.w . . . 4 π‘Š = (𝑁 WWalksN 𝐺)
39 wwlksn 29355 . . . . 5 (𝑁 ∈ β„•0 β†’ (𝑁 WWalksN 𝐺) = {π‘ž ∈ (WWalksβ€˜πΊ) ∣ (β™―β€˜π‘ž) = (𝑁 + 1)})
4039adantl 481 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ (𝑁 WWalksN 𝐺) = {π‘ž ∈ (WWalksβ€˜πΊ) ∣ (β™―β€˜π‘ž) = (𝑁 + 1)})
4138, 40eqtrid 2783 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ π‘Š = {π‘ž ∈ (WWalksβ€˜πΊ) ∣ (β™―β€˜π‘ž) = (𝑁 + 1)})
4236, 37, 41f1oeq123d 6828 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ (𝐹:𝑇–1-1-ontoβ†’π‘Š ↔ ((𝑝 ∈ (Walksβ€˜πΊ) ↦ (2nd β€˜π‘)) β†Ύ {𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}):{𝑝 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘)) = 𝑁}–1-1-ontoβ†’{π‘ž ∈ (WWalksβ€˜πΊ) ∣ (β™―β€˜π‘ž) = (𝑁 + 1)}))
4324, 42mpbird 256 1 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ 𝐹:𝑇–1-1-ontoβ†’π‘Š)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  {crab 3431   βŠ† wss 3949   class class class wbr 5149   ↦ cmpt 5232   β†Ύ cres 5679  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  (class class class)co 7412  1st c1st 7976  2nd c2nd 7977  β„‚cc 11111  1c1 11114   + caddc 11116  β„•0cn0 12477  β™―chash 14295  USPGraphcuspgr 28672  Walkscwlks 29117  WWalkscwwlks 29343   WWalksN cwwlksn 29344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1061  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-2o 8470  df-oadd 8473  df-er 8706  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-dju 9899  df-card 9937  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-n0 12478  df-xnn0 12550  df-z 12564  df-uz 12828  df-fz 13490  df-fzo 13633  df-hash 14296  df-word 14470  df-edg 28572  df-uhgr 28582  df-upgr 28606  df-uspgr 28674  df-wlks 29120  df-wwlks 29348  df-wwlksn 29349
This theorem is referenced by:  wlknwwlksnen  29407  wlksnwwlknvbij  29426
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