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Theorem wwlksnextprop 29163
Description: Adding additional properties to the set of walks (as words) of a fixed length starting at a fixed vertex. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 29-Oct-2022.)
Hypotheses
Ref Expression
wwlksnextprop.x 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
wwlksnextprop.e 𝐸 = (Edgβ€˜πΊ)
wwlksnextprop.y π‘Œ = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}
Assertion
Ref Expression
wwlksnextprop (𝑁 ∈ β„•0 β†’ {π‘₯ ∈ 𝑋 ∣ (π‘₯β€˜0) = 𝑃} = {π‘₯ ∈ 𝑋 ∣ βˆƒπ‘¦ ∈ π‘Œ ((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸)})
Distinct variable groups:   𝑀,𝐺   𝑀,𝑁   𝑀,𝑃   𝑦,𝐸   π‘₯,𝑁,𝑦   𝑦,𝑃   𝑦,𝑋   𝑦,π‘Œ   π‘₯,𝑀
Allowed substitution hints:   𝑃(π‘₯)   𝐸(π‘₯,𝑀)   𝐺(π‘₯,𝑦)   𝑋(π‘₯,𝑀)   π‘Œ(π‘₯,𝑀)

Proof of Theorem wwlksnextprop
StepHypRef Expression
1 eqidd 2733 . . . . 5 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ (π‘₯ prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)))
2 wwlksnextprop.x . . . . . . . . 9 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
32wwlksnextproplem1 29160 . . . . . . . 8 ((π‘₯ ∈ 𝑋 ∧ 𝑁 ∈ β„•0) β†’ ((π‘₯ prefix (𝑁 + 1))β€˜0) = (π‘₯β€˜0))
43ancoms 459 . . . . . . 7 ((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯ prefix (𝑁 + 1))β€˜0) = (π‘₯β€˜0))
54adantr 481 . . . . . 6 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ ((π‘₯ prefix (𝑁 + 1))β€˜0) = (π‘₯β€˜0))
6 eqeq2 2744 . . . . . . 7 ((π‘₯β€˜0) = 𝑃 β†’ (((π‘₯ prefix (𝑁 + 1))β€˜0) = (π‘₯β€˜0) ↔ ((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃))
76adantl 482 . . . . . 6 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ (((π‘₯ prefix (𝑁 + 1))β€˜0) = (π‘₯β€˜0) ↔ ((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃))
85, 7mpbid 231 . . . . 5 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ ((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃)
9 wwlksnextprop.e . . . . . . . 8 𝐸 = (Edgβ€˜πΊ)
102, 9wwlksnextproplem2 29161 . . . . . . 7 ((π‘₯ ∈ 𝑋 ∧ 𝑁 ∈ β„•0) β†’ {(lastSβ€˜(π‘₯ prefix (𝑁 + 1))), (lastSβ€˜π‘₯)} ∈ 𝐸)
1110ancoms 459 . . . . . 6 ((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) β†’ {(lastSβ€˜(π‘₯ prefix (𝑁 + 1))), (lastSβ€˜π‘₯)} ∈ 𝐸)
1211adantr 481 . . . . 5 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ {(lastSβ€˜(π‘₯ prefix (𝑁 + 1))), (lastSβ€˜π‘₯)} ∈ 𝐸)
13 simpr 485 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
1413adantr 481 . . . . . . 7 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ π‘₯ ∈ 𝑋)
15 simpr 485 . . . . . . 7 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ (π‘₯β€˜0) = 𝑃)
16 simpll 765 . . . . . . 7 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ 𝑁 ∈ β„•0)
17 wwlksnextprop.y . . . . . . . 8 π‘Œ = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}
182, 9, 17wwlksnextproplem3 29162 . . . . . . 7 ((π‘₯ ∈ 𝑋 ∧ (π‘₯β€˜0) = 𝑃 ∧ 𝑁 ∈ β„•0) β†’ (π‘₯ prefix (𝑁 + 1)) ∈ π‘Œ)
1914, 15, 16, 18syl3anc 1371 . . . . . 6 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ (π‘₯ prefix (𝑁 + 1)) ∈ π‘Œ)
20 eqeq2 2744 . . . . . . . 8 (𝑦 = (π‘₯ prefix (𝑁 + 1)) β†’ ((π‘₯ prefix (𝑁 + 1)) = 𝑦 ↔ (π‘₯ prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1))))
21 fveq1 6890 . . . . . . . . 9 (𝑦 = (π‘₯ prefix (𝑁 + 1)) β†’ (π‘¦β€˜0) = ((π‘₯ prefix (𝑁 + 1))β€˜0))
2221eqeq1d 2734 . . . . . . . 8 (𝑦 = (π‘₯ prefix (𝑁 + 1)) β†’ ((π‘¦β€˜0) = 𝑃 ↔ ((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃))
23 fveq2 6891 . . . . . . . . . 10 (𝑦 = (π‘₯ prefix (𝑁 + 1)) β†’ (lastSβ€˜π‘¦) = (lastSβ€˜(π‘₯ prefix (𝑁 + 1))))
2423preq1d 4743 . . . . . . . . 9 (𝑦 = (π‘₯ prefix (𝑁 + 1)) β†’ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} = {(lastSβ€˜(π‘₯ prefix (𝑁 + 1))), (lastSβ€˜π‘₯)})
2524eleq1d 2818 . . . . . . . 8 (𝑦 = (π‘₯ prefix (𝑁 + 1)) β†’ ({(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸 ↔ {(lastSβ€˜(π‘₯ prefix (𝑁 + 1))), (lastSβ€˜π‘₯)} ∈ 𝐸))
2620, 22, 253anbi123d 1436 . . . . . . 7 (𝑦 = (π‘₯ prefix (𝑁 + 1)) β†’ (((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸) ↔ ((π‘₯ prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)) ∧ ((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃 ∧ {(lastSβ€˜(π‘₯ prefix (𝑁 + 1))), (lastSβ€˜π‘₯)} ∈ 𝐸)))
2726adantl 482 . . . . . 6 ((((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) ∧ 𝑦 = (π‘₯ prefix (𝑁 + 1))) β†’ (((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸) ↔ ((π‘₯ prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)) ∧ ((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃 ∧ {(lastSβ€˜(π‘₯ prefix (𝑁 + 1))), (lastSβ€˜π‘₯)} ∈ 𝐸)))
2819, 27rspcedv 3605 . . . . 5 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ (((π‘₯ prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)) ∧ ((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃 ∧ {(lastSβ€˜(π‘₯ prefix (𝑁 + 1))), (lastSβ€˜π‘₯)} ∈ 𝐸) β†’ βˆƒπ‘¦ ∈ π‘Œ ((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸)))
291, 8, 12, 28mp3and 1464 . . . 4 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ βˆƒπ‘¦ ∈ π‘Œ ((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸))
3029ex 413 . . 3 ((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯β€˜0) = 𝑃 β†’ βˆƒπ‘¦ ∈ π‘Œ ((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸)))
3121eqcoms 2740 . . . . . . . . 9 ((π‘₯ prefix (𝑁 + 1)) = 𝑦 β†’ (π‘¦β€˜0) = ((π‘₯ prefix (𝑁 + 1))β€˜0))
3231eqeq1d 2734 . . . . . . . 8 ((π‘₯ prefix (𝑁 + 1)) = 𝑦 β†’ ((π‘¦β€˜0) = 𝑃 ↔ ((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃))
333eqcomd 2738 . . . . . . . . . . 11 ((π‘₯ ∈ 𝑋 ∧ 𝑁 ∈ β„•0) β†’ (π‘₯β€˜0) = ((π‘₯ prefix (𝑁 + 1))β€˜0))
3433ancoms 459 . . . . . . . . . 10 ((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯β€˜0) = ((π‘₯ prefix (𝑁 + 1))β€˜0))
3534adantr 481 . . . . . . . . 9 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (π‘₯β€˜0) = ((π‘₯ prefix (𝑁 + 1))β€˜0))
36 eqeq2 2744 . . . . . . . . . 10 (𝑃 = ((π‘₯ prefix (𝑁 + 1))β€˜0) β†’ ((π‘₯β€˜0) = 𝑃 ↔ (π‘₯β€˜0) = ((π‘₯ prefix (𝑁 + 1))β€˜0)))
3736eqcoms 2740 . . . . . . . . 9 (((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃 β†’ ((π‘₯β€˜0) = 𝑃 ↔ (π‘₯β€˜0) = ((π‘₯ prefix (𝑁 + 1))β€˜0)))
3835, 37imbitrrid 245 . . . . . . . 8 (((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃 β†’ (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (π‘₯β€˜0) = 𝑃))
3932, 38syl6bi 252 . . . . . . 7 ((π‘₯ prefix (𝑁 + 1)) = 𝑦 β†’ ((π‘¦β€˜0) = 𝑃 β†’ (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (π‘₯β€˜0) = 𝑃)))
4039imp 407 . . . . . 6 (((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃) β†’ (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (π‘₯β€˜0) = 𝑃))
41403adant3 1132 . . . . 5 (((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸) β†’ (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (π‘₯β€˜0) = 𝑃))
4241com12 32 . . . 4 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸) β†’ (π‘₯β€˜0) = 𝑃))
4342rexlimdva 3155 . . 3 ((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) β†’ (βˆƒπ‘¦ ∈ π‘Œ ((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸) β†’ (π‘₯β€˜0) = 𝑃))
4430, 43impbid 211 . 2 ((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯β€˜0) = 𝑃 ↔ βˆƒπ‘¦ ∈ π‘Œ ((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸)))
4544rabbidva 3439 1 (𝑁 ∈ β„•0 β†’ {π‘₯ ∈ 𝑋 ∣ (π‘₯β€˜0) = 𝑃} = {π‘₯ ∈ 𝑋 ∣ βˆƒπ‘¦ ∈ π‘Œ ((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  {crab 3432  {cpr 4630  β€˜cfv 6543  (class class class)co 7408  0cc0 11109  1c1 11110   + caddc 11112  β„•0cn0 12471  lastSclsw 14511   prefix cpfx 14619  Edgcedg 28304   WWalksN cwwlksn 29077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-fzo 13627  df-hash 14290  df-word 14464  df-lsw 14512  df-substr 14590  df-pfx 14620  df-wwlks 29081  df-wwlksn 29082
This theorem is referenced by: (None)
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