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Theorem wwlkextprop 26272
 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.)
Hypotheses
Ref Expression
wwlkextprop.x 𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))
wwlkextprop.y 𝑌 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}
Assertion
Ref Expression
wwlkextprop (𝑁 ∈ ℕ0 → {𝑥𝑋 ∣ (𝑥‘0) = 𝑃} = {𝑥𝑋 ∣ ∃𝑦𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)})
Distinct variable groups:   𝑤,𝐸   𝑤,𝑁   𝑤,𝑃   𝑤,𝑉   𝑦,𝐸   𝑥,𝑁,𝑦,𝑤   𝑦,𝑃   𝑦,𝑋   𝑦,𝑌
Allowed substitution hints:   𝑃(𝑥)   𝐸(𝑥)   𝑉(𝑥,𝑦)   𝑋(𝑥,𝑤)   𝑌(𝑥,𝑤)

Proof of Theorem wwlkextprop
StepHypRef Expression
1 eqidd 2611 . . . . 5 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩))
2 wwlkextprop.x . . . . . . . . 9 𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))
32wwlkextproplem1 26269 . . . . . . . 8 ((𝑥𝑋𝑁 ∈ ℕ0) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑥‘0))
43ancoms 468 . . . . . . 7 ((𝑁 ∈ ℕ0𝑥𝑋) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑥‘0))
54adantr 480 . . . . . 6 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑥‘0))
6 eqeq2 2621 . . . . . . 7 ((𝑥‘0) = 𝑃 → (((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑥‘0) ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃))
76adantl 481 . . . . . 6 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑥‘0) ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃))
85, 7mpbid 221 . . . . 5 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃)
92wwlkextproplem2 26270 . . . . . . 7 ((𝑥𝑋𝑁 ∈ ℕ0) → {( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑥)} ∈ ran 𝐸)
109ancoms 468 . . . . . 6 ((𝑁 ∈ ℕ0𝑥𝑋) → {( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑥)} ∈ ran 𝐸)
1110adantr 480 . . . . 5 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → {( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑥)} ∈ ran 𝐸)
12 simpr 476 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑥𝑋) → 𝑥𝑋)
1312adantr 480 . . . . . . 7 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → 𝑥𝑋)
14 simpr 476 . . . . . . 7 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥‘0) = 𝑃)
15 simpll 786 . . . . . . 7 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → 𝑁 ∈ ℕ0)
16 wwlkextprop.y . . . . . . . 8 𝑌 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}
172, 16wwlkextproplem3 26271 . . . . . . 7 ((𝑥𝑋 ∧ (𝑥‘0) = 𝑃𝑁 ∈ ℕ0) → (𝑥 substr ⟨0, (𝑁 + 1)⟩) ∈ 𝑌)
1813, 14, 15, 17syl3anc 1318 . . . . . 6 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥 substr ⟨0, (𝑁 + 1)⟩) ∈ 𝑌)
19 eqeq2 2621 . . . . . . . 8 (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ↔ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩)))
20 fveq1 6102 . . . . . . . . 9 (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → (𝑦‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0))
2120eqeq1d 2612 . . . . . . . 8 (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → ((𝑦‘0) = 𝑃 ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃))
22 fveq2 6103 . . . . . . . . . 10 (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → ( lastS ‘𝑦) = ( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)))
2322preq1d 4218 . . . . . . . . 9 (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → {( lastS ‘𝑦), ( lastS ‘𝑥)} = {( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑥)})
2423eleq1d 2672 . . . . . . . 8 (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → ({( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸 ↔ {( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑥)} ∈ ran 𝐸))
2519, 21, 243anbi123d 1391 . . . . . . 7 (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸) ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) ∧ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃 ∧ {( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑥)} ∈ ran 𝐸)))
2625adantl 481 . . . . . 6 ((((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) ∧ 𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩)) → (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸) ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) ∧ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃 ∧ {( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑥)} ∈ ran 𝐸)))
2718, 26rspcedv 3286 . . . . 5 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) ∧ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃 ∧ {( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑥)} ∈ ran 𝐸) → ∃𝑦𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)))
281, 8, 11, 27mp3and 1419 . . . 4 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → ∃𝑦𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸))
2928ex 449 . . 3 ((𝑁 ∈ ℕ0𝑥𝑋) → ((𝑥‘0) = 𝑃 → ∃𝑦𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)))
3020eqcoms 2618 . . . . . . . . 9 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 → (𝑦‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0))
3130eqeq1d 2612 . . . . . . . 8 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 → ((𝑦‘0) = 𝑃 ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃))
323eqcomd 2616 . . . . . . . . . . 11 ((𝑥𝑋𝑁 ∈ ℕ0) → (𝑥‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0))
3332ancoms 468 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑥𝑋) → (𝑥‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0))
3433adantr 480 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0))
35 eqeq2 2621 . . . . . . . . . 10 (𝑃 = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) → ((𝑥‘0) = 𝑃 ↔ (𝑥‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0)))
3635eqcoms 2618 . . . . . . . . 9 (((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃 → ((𝑥‘0) = 𝑃 ↔ (𝑥‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0)))
3734, 36syl5ibr 235 . . . . . . . 8 (((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃 → (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = 𝑃))
3831, 37syl6bi 242 . . . . . . 7 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 → ((𝑦‘0) = 𝑃 → (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = 𝑃)))
3938imp 444 . . . . . 6 (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) → (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = 𝑃))
40393adant3 1074 . . . . 5 (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸) → (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = 𝑃))
4140com12 32 . . . 4 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸) → (𝑥‘0) = 𝑃))
4241rexlimdva 3013 . . 3 ((𝑁 ∈ ℕ0𝑥𝑋) → (∃𝑦𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸) → (𝑥‘0) = 𝑃))
4329, 42impbid 201 . 2 ((𝑁 ∈ ℕ0𝑥𝑋) → ((𝑥‘0) = 𝑃 ↔ ∃𝑦𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)))
4443rabbidva 3163 1 (𝑁 ∈ ℕ0 → {𝑥𝑋 ∣ (𝑥‘0) = 𝑃} = {𝑥𝑋 ∣ ∃𝑦𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  {cpr 4127  ⟨cop 4131  ran crn 5039  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  ℕ0cn0 11169   lastS clsw 13147   substr csubstr 13150   WWalksN cwwlkn 26206 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-substr 13158  df-wwlk 26207  df-wwlkn 26208 This theorem is referenced by: (None)
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