Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ewlksfval Structured version   Visualization version   GIF version

Theorem ewlksfval 40803
 Description: The set of s-walks of edges (in a hypergraph). (Contributed by AV, 4-Jan-2021.)
Hypothesis
Ref Expression
ewlksfval.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
ewlksfval ((𝐺𝑊𝑆 ∈ ℕ0*) → (𝐺 EdgWalks 𝑆) = {𝑓 ∣ (𝑓 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓𝑘)))))})
Distinct variable groups:   𝑓,𝐺,𝑘   𝑆,𝑓,𝑘   𝑓,𝑊,𝑘
Allowed substitution hints:   𝐼(𝑓,𝑘)

Proof of Theorem ewlksfval
Dummy variables 𝑔 𝑖 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ewlks 40799 . . . 4 EdgWalks = (𝑔 ∈ V, 𝑠 ∈ ℕ0* ↦ {𝑓[(iEdg‘𝑔) / 𝑖](𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑠 ≤ (#‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓𝑘)))))})
21a1i 11 . . 3 ((𝐺𝑊𝑆 ∈ ℕ0*) → EdgWalks = (𝑔 ∈ V, 𝑠 ∈ ℕ0* ↦ {𝑓[(iEdg‘𝑔) / 𝑖](𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑠 ≤ (#‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓𝑘)))))}))
3 fvex 6113 . . . . . . 7 (iEdg‘𝑔) ∈ V
43a1i 11 . . . . . 6 ((𝑔 = 𝐺𝑠 = 𝑆) → (iEdg‘𝑔) ∈ V)
5 simpr 476 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑖 = (iEdg‘𝑔))
6 fveq2 6103 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
76adantr 480 . . . . . . . . . . . 12 ((𝑔 = 𝐺𝑠 = 𝑆) → (iEdg‘𝑔) = (iEdg‘𝐺))
87adantr 480 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (iEdg‘𝑔) = (iEdg‘𝐺))
95, 8eqtrd 2644 . . . . . . . . . 10 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑖 = (iEdg‘𝐺))
109dmeqd 5248 . . . . . . . . 9 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → dom 𝑖 = dom (iEdg‘𝐺))
11 wrdeq 13182 . . . . . . . . 9 (dom 𝑖 = dom (iEdg‘𝐺) → Word dom 𝑖 = Word dom (iEdg‘𝐺))
1210, 11syl 17 . . . . . . . 8 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → Word dom 𝑖 = Word dom (iEdg‘𝐺))
1312eleq2d 2673 . . . . . . 7 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑓 ∈ Word dom 𝑖𝑓 ∈ Word dom (iEdg‘𝐺)))
14 simpr 476 . . . . . . . . . 10 ((𝑔 = 𝐺𝑠 = 𝑆) → 𝑠 = 𝑆)
1514adantr 480 . . . . . . . . 9 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑠 = 𝑆)
169fveq1d 6105 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑖‘(𝑓‘(𝑘 − 1))) = ((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))))
179fveq1d 6105 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑖‘(𝑓𝑘)) = ((iEdg‘𝐺)‘(𝑓𝑘)))
1816, 17ineq12d 3777 . . . . . . . . . 10 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → ((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓𝑘))) = (((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘))))
1918fveq2d 6107 . . . . . . . . 9 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (#‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓𝑘)))) = (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘)))))
2015, 19breq12d 4596 . . . . . . . 8 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑠 ≤ (#‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓𝑘)))) ↔ 𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘))))))
2120ralbidv 2969 . . . . . . 7 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (∀𝑘 ∈ (1..^(#‘𝑓))𝑠 ≤ (#‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓𝑘)))) ↔ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘))))))
2213, 21anbi12d 743 . . . . . 6 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → ((𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑠 ≤ (#‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓𝑘))))) ↔ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘)))))))
234, 22sbcied 3439 . . . . 5 ((𝑔 = 𝐺𝑠 = 𝑆) → ([(iEdg‘𝑔) / 𝑖](𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑠 ≤ (#‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓𝑘))))) ↔ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘)))))))
2423abbidv 2728 . . . 4 ((𝑔 = 𝐺𝑠 = 𝑆) → {𝑓[(iEdg‘𝑔) / 𝑖](𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑠 ≤ (#‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓𝑘)))))} = {𝑓 ∣ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘)))))})
2524adantl 481 . . 3 (((𝐺𝑊𝑆 ∈ ℕ0*) ∧ (𝑔 = 𝐺𝑠 = 𝑆)) → {𝑓[(iEdg‘𝑔) / 𝑖](𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑠 ≤ (#‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓𝑘)))))} = {𝑓 ∣ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘)))))})
26 elex 3185 . . . 4 (𝐺𝑊𝐺 ∈ V)
2726adantr 480 . . 3 ((𝐺𝑊𝑆 ∈ ℕ0*) → 𝐺 ∈ V)
28 simpr 476 . . 3 ((𝐺𝑊𝑆 ∈ ℕ0*) → 𝑆 ∈ ℕ0*)
29 df-rab 2905 . . . 4 {𝑓 ∈ Word dom (iEdg‘𝐺) ∣ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘))))} = {𝑓 ∣ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘)))))}
30 fvex 6113 . . . . . . . 8 (iEdg‘𝐺) ∈ V
3130dmex 6991 . . . . . . 7 dom (iEdg‘𝐺) ∈ V
3231wrdexi 13172 . . . . . 6 Word dom (iEdg‘𝐺) ∈ V
3332rabex 4740 . . . . 5 {𝑓 ∈ Word dom (iEdg‘𝐺) ∣ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘))))} ∈ V
3433a1i 11 . . . 4 ((𝐺𝑊𝑆 ∈ ℕ0*) → {𝑓 ∈ Word dom (iEdg‘𝐺) ∣ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘))))} ∈ V)
3529, 34syl5eqelr 2693 . . 3 ((𝐺𝑊𝑆 ∈ ℕ0*) → {𝑓 ∣ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘)))))} ∈ V)
362, 25, 27, 28, 35ovmpt2d 6686 . 2 ((𝐺𝑊𝑆 ∈ ℕ0*) → (𝐺 EdgWalks 𝑆) = {𝑓 ∣ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘)))))})
37 ewlksfval.i . . . . . . . . 9 𝐼 = (iEdg‘𝐺)
3837eqcomi 2619 . . . . . . . 8 (iEdg‘𝐺) = 𝐼
3938a1i 11 . . . . . . 7 ((𝐺𝑊𝑆 ∈ ℕ0*) → (iEdg‘𝐺) = 𝐼)
4039dmeqd 5248 . . . . . 6 ((𝐺𝑊𝑆 ∈ ℕ0*) → dom (iEdg‘𝐺) = dom 𝐼)
41 wrdeq 13182 . . . . . 6 (dom (iEdg‘𝐺) = dom 𝐼 → Word dom (iEdg‘𝐺) = Word dom 𝐼)
4240, 41syl 17 . . . . 5 ((𝐺𝑊𝑆 ∈ ℕ0*) → Word dom (iEdg‘𝐺) = Word dom 𝐼)
4342eleq2d 2673 . . . 4 ((𝐺𝑊𝑆 ∈ ℕ0*) → (𝑓 ∈ Word dom (iEdg‘𝐺) ↔ 𝑓 ∈ Word dom 𝐼))
4439fveq1d 6105 . . . . . . . 8 ((𝐺𝑊𝑆 ∈ ℕ0*) → ((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) = (𝐼‘(𝑓‘(𝑘 − 1))))
4539fveq1d 6105 . . . . . . . 8 ((𝐺𝑊𝑆 ∈ ℕ0*) → ((iEdg‘𝐺)‘(𝑓𝑘)) = (𝐼‘(𝑓𝑘)))
4644, 45ineq12d 3777 . . . . . . 7 ((𝐺𝑊𝑆 ∈ ℕ0*) → (((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘))) = ((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓𝑘))))
4746fveq2d 6107 . . . . . 6 ((𝐺𝑊𝑆 ∈ ℕ0*) → (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘)))) = (#‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓𝑘)))))
4847breq2d 4595 . . . . 5 ((𝐺𝑊𝑆 ∈ ℕ0*) → (𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘)))) ↔ 𝑆 ≤ (#‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓𝑘))))))
4948ralbidv 2969 . . . 4 ((𝐺𝑊𝑆 ∈ ℕ0*) → (∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘)))) ↔ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓𝑘))))))
5043, 49anbi12d 743 . . 3 ((𝐺𝑊𝑆 ∈ ℕ0*) → ((𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘))))) ↔ (𝑓 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓𝑘)))))))
5150abbidv 2728 . 2 ((𝐺𝑊𝑆 ∈ ℕ0*) → {𝑓 ∣ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘)))))} = {𝑓 ∣ (𝑓 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓𝑘)))))})
5236, 51eqtrd 2644 1 ((𝐺𝑊𝑆 ∈ ℕ0*) → (𝐺 EdgWalks 𝑆) = {𝑓 ∣ (𝑓 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓𝑘)))))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  {crab 2900  Vcvv 3173  [wsbc 3402   ∩ cin 3539   class class class wbr 4583  dom cdm 5038  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1c1 9816   ≤ cle 9954   − cmin 10145  ℕ0*cxnn0 11240  ..^cfzo 12334  #chash 12979  Word cword 13146  iEdgciedg 25674   EdgWalks cewlks 40795 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-ewlks 40799 This theorem is referenced by:  isewlk  40804
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