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Theorem wwlksnextprop 29795
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 = (Edg‘𝐺)
wwlksnextprop.y 𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
Assertion
Ref Expression
wwlksnextprop (𝑁 ∈ ℕ0 → {𝑥𝑋 ∣ (𝑥‘0) = 𝑃} = {𝑥𝑋 ∣ ∃𝑦𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)})
Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑤,𝑃   𝑦,𝐸   𝑥,𝑁,𝑦   𝑦,𝑃   𝑦,𝑋   𝑦,𝑌   𝑥,𝑤
Allowed substitution hints:   𝑃(𝑥)   𝐸(𝑥,𝑤)   𝐺(𝑥,𝑦)   𝑋(𝑥,𝑤)   𝑌(𝑥,𝑤)

Proof of Theorem wwlksnextprop
StepHypRef Expression
1 eqidd 2726 . . . . 5 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)))
2 wwlksnextprop.x . . . . . . . . 9 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
32wwlksnextproplem1 29792 . . . . . . . 8 ((𝑥𝑋𝑁 ∈ ℕ0) → ((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0))
43ancoms 457 . . . . . . 7 ((𝑁 ∈ ℕ0𝑥𝑋) → ((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0))
54adantr 479 . . . . . 6 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → ((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0))
6 eqeq2 2737 . . . . . . 7 ((𝑥‘0) = 𝑃 → (((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0) ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
76adantl 480 . . . . . 6 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0) ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
85, 7mpbid 231 . . . . 5 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃)
9 wwlksnextprop.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
102, 9wwlksnextproplem2 29793 . . . . . . 7 ((𝑥𝑋𝑁 ∈ ℕ0) → {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)
1110ancoms 457 . . . . . 6 ((𝑁 ∈ ℕ0𝑥𝑋) → {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)
1211adantr 479 . . . . 5 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)
13 simpr 483 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑥𝑋) → 𝑥𝑋)
1413adantr 479 . . . . . . 7 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → 𝑥𝑋)
15 simpr 483 . . . . . . 7 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥‘0) = 𝑃)
16 simpll 765 . . . . . . 7 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → 𝑁 ∈ ℕ0)
17 wwlksnextprop.y . . . . . . . 8 𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
182, 9, 17wwlksnextproplem3 29794 . . . . . . 7 ((𝑥𝑋 ∧ (𝑥‘0) = 𝑃𝑁 ∈ ℕ0) → (𝑥 prefix (𝑁 + 1)) ∈ 𝑌)
1914, 15, 16, 18syl3anc 1368 . . . . . 6 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥 prefix (𝑁 + 1)) ∈ 𝑌)
20 eqeq2 2737 . . . . . . . 8 (𝑦 = (𝑥 prefix (𝑁 + 1)) → ((𝑥 prefix (𝑁 + 1)) = 𝑦 ↔ (𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1))))
21 fveq1 6895 . . . . . . . . 9 (𝑦 = (𝑥 prefix (𝑁 + 1)) → (𝑦‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
2221eqeq1d 2727 . . . . . . . 8 (𝑦 = (𝑥 prefix (𝑁 + 1)) → ((𝑦‘0) = 𝑃 ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
23 fveq2 6896 . . . . . . . . . 10 (𝑦 = (𝑥 prefix (𝑁 + 1)) → (lastS‘𝑦) = (lastS‘(𝑥 prefix (𝑁 + 1))))
2423preq1d 4745 . . . . . . . . 9 (𝑦 = (𝑥 prefix (𝑁 + 1)) → {(lastS‘𝑦), (lastS‘𝑥)} = {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)})
2524eleq1d 2810 . . . . . . . 8 (𝑦 = (𝑥 prefix (𝑁 + 1)) → ({(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸 ↔ {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸))
2620, 22, 253anbi123d 1432 . . . . . . 7 (𝑦 = (𝑥 prefix (𝑁 + 1)) → (((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) ↔ ((𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) ∧ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 ∧ {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)))
2726adantl 480 . . . . . 6 ((((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) ∧ 𝑦 = (𝑥 prefix (𝑁 + 1))) → (((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) ↔ ((𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) ∧ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 ∧ {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)))
2819, 27rspcedv 3599 . . . . 5 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (((𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) ∧ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 ∧ {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸) → ∃𝑦𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)))
291, 8, 12, 28mp3and 1460 . . . 4 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → ∃𝑦𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸))
3029ex 411 . . 3 ((𝑁 ∈ ℕ0𝑥𝑋) → ((𝑥‘0) = 𝑃 → ∃𝑦𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)))
3121eqcoms 2733 . . . . . . . . 9 ((𝑥 prefix (𝑁 + 1)) = 𝑦 → (𝑦‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
3231eqeq1d 2727 . . . . . . . 8 ((𝑥 prefix (𝑁 + 1)) = 𝑦 → ((𝑦‘0) = 𝑃 ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
333eqcomd 2731 . . . . . . . . . . 11 ((𝑥𝑋𝑁 ∈ ℕ0) → (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
3433ancoms 457 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑥𝑋) → (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
3534adantr 479 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
36 eqeq2 2737 . . . . . . . . . 10 (𝑃 = ((𝑥 prefix (𝑁 + 1))‘0) → ((𝑥‘0) = 𝑃 ↔ (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0)))
3736eqcoms 2733 . . . . . . . . 9 (((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 → ((𝑥‘0) = 𝑃 ↔ (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0)))
3835, 37imbitrrid 245 . . . . . . . 8 (((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 → (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = 𝑃))
3932, 38biimtrdi 252 . . . . . . 7 ((𝑥 prefix (𝑁 + 1)) = 𝑦 → ((𝑦‘0) = 𝑃 → (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = 𝑃)))
4039imp 405 . . . . . 6 (((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃) → (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = 𝑃))
41403adant3 1129 . . . . 5 (((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = 𝑃))
4241com12 32 . . . 4 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥‘0) = 𝑃))
4342rexlimdva 3144 . . 3 ((𝑁 ∈ ℕ0𝑥𝑋) → (∃𝑦𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥‘0) = 𝑃))
4430, 43impbid 211 . 2 ((𝑁 ∈ ℕ0𝑥𝑋) → ((𝑥‘0) = 𝑃 ↔ ∃𝑦𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)))
4544rabbidva 3425 1 (𝑁 ∈ ℕ0 → {𝑥𝑋 ∣ (𝑥‘0) = 𝑃} = {𝑥𝑋 ∣ ∃𝑦𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wrex 3059  {crab 3418  {cpr 4632  cfv 6549  (class class class)co 7419  0cc0 11140  1c1 11141   + caddc 11143  0cn0 12505  lastSclsw 14548   prefix cpfx 14656  Edgcedg 28932   WWalksN cwwlksn 29709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-n0 12506  df-z 12592  df-uz 12856  df-fz 13520  df-fzo 13663  df-hash 14326  df-word 14501  df-lsw 14549  df-substr 14627  df-pfx 14657  df-wwlks 29713  df-wwlksn 29714
This theorem is referenced by: (None)
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