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Theorem wwlknprop 26214
 Description: Properties of a walk of a fixed length (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 16-Jul-2018.)
Assertion
Ref Expression
wwlknprop (𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0𝑃 ∈ Word 𝑉)))

Proof of Theorem wwlknprop
Dummy variables 𝑒 𝑛 𝑣 𝑤 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wwlkn 26208 . . 3 WWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)}))
2 df-rab 2905 . . . . . . 7 {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)} = {𝑤 ∣ (𝑤 ∈ (𝑣 WWalks 𝑒) ∧ (#‘𝑤) = (𝑛 + 1))}
3 iswwlk 26211 . . . . . . . . . 10 ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (𝑤 ∈ (𝑣 WWalks 𝑒) ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑣 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒)))
43adantr 480 . . . . . . . . 9 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ 𝑛 ∈ ℕ0) → (𝑤 ∈ (𝑣 WWalks 𝑒) ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑣 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒)))
54anbi1d 737 . . . . . . . 8 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ 𝑛 ∈ ℕ0) → ((𝑤 ∈ (𝑣 WWalks 𝑒) ∧ (#‘𝑤) = (𝑛 + 1)) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑣 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ∧ (#‘𝑤) = (𝑛 + 1))))
65abbidv 2728 . . . . . . 7 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ 𝑛 ∈ ℕ0) → {𝑤 ∣ (𝑤 ∈ (𝑣 WWalks 𝑒) ∧ (#‘𝑤) = (𝑛 + 1))} = {𝑤 ∣ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑣 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ∧ (#‘𝑤) = (𝑛 + 1))})
72, 6syl5eq 2656 . . . . . 6 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ 𝑛 ∈ ℕ0) → {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)} = {𝑤 ∣ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑣 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ∧ (#‘𝑤) = (𝑛 + 1))})
8 3anan12 1044 . . . . . . . . . 10 ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑣 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ↔ (𝑤 ∈ Word 𝑣 ∧ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒)))
98anbi1i 727 . . . . . . . . 9 (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑣 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ∧ (#‘𝑤) = (𝑛 + 1)) ↔ ((𝑤 ∈ Word 𝑣 ∧ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒)) ∧ (#‘𝑤) = (𝑛 + 1)))
10 anass 679 . . . . . . . . 9 (((𝑤 ∈ Word 𝑣 ∧ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒)) ∧ (#‘𝑤) = (𝑛 + 1)) ↔ (𝑤 ∈ Word 𝑣 ∧ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ∧ (#‘𝑤) = (𝑛 + 1))))
119, 10bitri 263 . . . . . . . 8 (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑣 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ∧ (#‘𝑤) = (𝑛 + 1)) ↔ (𝑤 ∈ Word 𝑣 ∧ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ∧ (#‘𝑤) = (𝑛 + 1))))
1211abbii 2726 . . . . . . 7 {𝑤 ∣ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑣 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ∧ (#‘𝑤) = (𝑛 + 1))} = {𝑤 ∣ (𝑤 ∈ Word 𝑣 ∧ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ∧ (#‘𝑤) = (𝑛 + 1)))}
13 df-rab 2905 . . . . . . 7 {𝑤 ∈ Word 𝑣 ∣ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ∧ (#‘𝑤) = (𝑛 + 1))} = {𝑤 ∣ (𝑤 ∈ Word 𝑣 ∧ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ∧ (#‘𝑤) = (𝑛 + 1)))}
1412, 13eqtr4i 2635 . . . . . 6 {𝑤 ∣ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑣 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ∧ (#‘𝑤) = (𝑛 + 1))} = {𝑤 ∈ Word 𝑣 ∣ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ∧ (#‘𝑤) = (𝑛 + 1))}
157, 14syl6eq 2660 . . . . 5 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ 𝑛 ∈ ℕ0) → {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)} = {𝑤 ∈ Word 𝑣 ∣ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ∧ (#‘𝑤) = (𝑛 + 1))})
1615mpteq2dva 4672 . . . 4 ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)}) = (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ Word 𝑣 ∣ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ∧ (#‘𝑤) = (𝑛 + 1))}))
1716mpt2eq3ia 6618 . . 3 (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)})) = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ Word 𝑣 ∣ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ∧ (#‘𝑤) = (𝑛 + 1))}))
181, 17eqtri 2632 . 2 WWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ Word 𝑣 ∣ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒) ∧ (#‘𝑤) = (𝑛 + 1))}))
1918elovmptnn0wrd 13203 1 (𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0𝑃 ∈ Word 𝑉)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173  ∅c0 3874  {cpr 4127   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ℕ0cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   WWalks cwwlk 26205   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-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-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-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-wwlk 26207  df-wwlkn 26208 This theorem is referenced by:  wwlknimp  26215  wwlkn0  26217  wlklniswwlkn2  26228  wwlkiswwlkn  26230  wwlknred  26251  wwlknext  26252  wwlkextwrd  26256  wwlkextsur  26259  wwlkextbij0  26260  wwlknndef  26265  wwlkextproplem3  26271  wwlkext2clwwlk  26331  numclwwlk2lem1  26629
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