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Theorem wlksnwwlknvbij 28895
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, 22-Jul-2018.) (Revised by AV, 5-Aug-2022.)
Assertion
Ref Expression
wlksnwwlknvbij ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ βˆƒπ‘“ 𝑓:{𝑝 ∈ (Walksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ ((2nd β€˜π‘)β€˜0) = 𝑋)}–1-1-ontoβ†’{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋})
Distinct variable groups:   𝑓,𝐺,𝑝,𝑀   𝑓,𝑁,𝑝,𝑀   𝑓,𝑋,𝑝,𝑀

Proof of Theorem wlksnwwlknvbij
Dummy variable π‘ž is distinct from all other variables.
StepHypRef Expression
1 fvex 6856 . . . . 5 (Walksβ€˜πΊ) ∈ V
21mptrabex 7176 . . . 4 (𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ↦ (2nd β€˜π‘)) ∈ V
32resex 5986 . . 3 ((𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ↦ (2nd β€˜π‘)) β†Ύ {𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋}) ∈ V
4 eqid 2733 . . . 4 (𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ↦ (2nd β€˜π‘)) = (𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ↦ (2nd β€˜π‘))
5 eqid 2733 . . . . 5 {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} = {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁}
6 eqid 2733 . . . . 5 (𝑁 WWalksN 𝐺) = (𝑁 WWalksN 𝐺)
75, 6, 4wlknwwlksnbij 28875 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ (𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ↦ (2nd β€˜π‘)):{π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁}–1-1-ontoβ†’(𝑁 WWalksN 𝐺))
8 fveq1 6842 . . . . . 6 (𝑀 = (2nd β€˜π‘) β†’ (π‘€β€˜0) = ((2nd β€˜π‘)β€˜0))
98eqeq1d 2735 . . . . 5 (𝑀 = (2nd β€˜π‘) β†’ ((π‘€β€˜0) = 𝑋 ↔ ((2nd β€˜π‘)β€˜0) = 𝑋))
1093ad2ant3 1136 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) ∧ 𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∧ 𝑀 = (2nd β€˜π‘)) β†’ ((π‘€β€˜0) = 𝑋 ↔ ((2nd β€˜π‘)β€˜0) = 𝑋))
114, 7, 10f1oresrab 7074 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ ((𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ↦ (2nd β€˜π‘)) β†Ύ {𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋}):{𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋}–1-1-ontoβ†’{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋})
12 f1oeq1 6773 . . . 4 (𝑓 = ((𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ↦ (2nd β€˜π‘)) β†Ύ {𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋}) β†’ (𝑓:{𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋}–1-1-ontoβ†’{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} ↔ ((𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ↦ (2nd β€˜π‘)) β†Ύ {𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋}):{𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋}–1-1-ontoβ†’{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}))
1312spcegv 3555 . . 3 (((𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ↦ (2nd β€˜π‘)) β†Ύ {𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋}) ∈ V β†’ (((𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ↦ (2nd β€˜π‘)) β†Ύ {𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋}):{𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋}–1-1-ontoβ†’{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} β†’ βˆƒπ‘“ 𝑓:{𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋}–1-1-ontoβ†’{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}))
143, 11, 13mpsyl 68 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ βˆƒπ‘“ 𝑓:{𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋}–1-1-ontoβ†’{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋})
15 2fveq3 6848 . . . . . . 7 (𝑝 = π‘ž β†’ (β™―β€˜(1st β€˜π‘)) = (β™―β€˜(1st β€˜π‘ž)))
1615eqeq1d 2735 . . . . . 6 (𝑝 = π‘ž β†’ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ↔ (β™―β€˜(1st β€˜π‘ž)) = 𝑁))
1716rabrabi 3424 . . . . 5 {𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋} = {𝑝 ∈ (Walksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ ((2nd β€˜π‘)β€˜0) = 𝑋)}
1817eqcomi 2742 . . . 4 {𝑝 ∈ (Walksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ ((2nd β€˜π‘)β€˜0) = 𝑋)} = {𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋}
19 f1oeq2 6774 . . . 4 ({𝑝 ∈ (Walksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ ((2nd β€˜π‘)β€˜0) = 𝑋)} = {𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋} β†’ (𝑓:{𝑝 ∈ (Walksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ ((2nd β€˜π‘)β€˜0) = 𝑋)}–1-1-ontoβ†’{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} ↔ 𝑓:{𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋}–1-1-ontoβ†’{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}))
2018, 19mp1i 13 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ (𝑓:{𝑝 ∈ (Walksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ ((2nd β€˜π‘)β€˜0) = 𝑋)}–1-1-ontoβ†’{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} ↔ 𝑓:{𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋}–1-1-ontoβ†’{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}))
2120exbidv 1925 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ (βˆƒπ‘“ 𝑓:{𝑝 ∈ (Walksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ ((2nd β€˜π‘)β€˜0) = 𝑋)}–1-1-ontoβ†’{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋} ↔ βˆƒπ‘“ 𝑓:{𝑝 ∈ {π‘ž ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘ž)) = 𝑁} ∣ ((2nd β€˜π‘)β€˜0) = 𝑋}–1-1-ontoβ†’{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋}))
2214, 21mpbird 257 1 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•0) β†’ βˆƒπ‘“ 𝑓:{𝑝 ∈ (Walksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ ((2nd β€˜π‘)β€˜0) = 𝑋)}–1-1-ontoβ†’{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑋})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {crab 3406  Vcvv 3444   ↦ cmpt 5189   β†Ύ cres 5636  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  0cc0 11056  β„•0cn0 12418  β™―chash 14236  USPGraphcuspgr 28141  Walkscwlks 28586   WWalksN cwwlksn 28813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-oadd 8417  df-er 8651  df-map 8770  df-pm 8771  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-dju 9842  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-n0 12419  df-xnn0 12491  df-z 12505  df-uz 12769  df-fz 13431  df-fzo 13574  df-hash 14237  df-word 14409  df-edg 28041  df-uhgr 28051  df-upgr 28075  df-uspgr 28143  df-wlks 28589  df-wwlks 28817  df-wwlksn 28818
This theorem is referenced by:  rusgrnumwlkg  28964
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