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Theorem ewlksfval 26367
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 26364 . . . 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 6158 . . . . . . 7 (iEdg‘𝑔) ∈ V
43a1i 11 . . . . . 6 ((𝑔 = 𝐺𝑠 = 𝑆) → (iEdg‘𝑔) ∈ V)
5 simpr 477 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑖 = (iEdg‘𝑔))
6 fveq2 6148 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
76adantr 481 . . . . . . . . . . . 12 ((𝑔 = 𝐺𝑠 = 𝑆) → (iEdg‘𝑔) = (iEdg‘𝐺))
87adantr 481 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (iEdg‘𝑔) = (iEdg‘𝐺))
95, 8eqtrd 2655 . . . . . . . . . 10 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑖 = (iEdg‘𝐺))
109dmeqd 5286 . . . . . . . . 9 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → dom 𝑖 = dom (iEdg‘𝐺))
11 wrdeq 13266 . . . . . . . . 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 2684 . . . . . . 7 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑓 ∈ Word dom 𝑖𝑓 ∈ Word dom (iEdg‘𝐺)))
14 simpr 477 . . . . . . . . . 10 ((𝑔 = 𝐺𝑠 = 𝑆) → 𝑠 = 𝑆)
1514adantr 481 . . . . . . . . 9 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑠 = 𝑆)
169fveq1d 6150 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑖‘(𝑓‘(𝑘 − 1))) = ((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))))
179fveq1d 6150 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑖‘(𝑓𝑘)) = ((iEdg‘𝐺)‘(𝑓𝑘)))
1816, 17ineq12d 3793 . . . . . . . . . 10 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → ((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓𝑘))) = (((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘))))
1918fveq2d 6152 . . . . . . . . 9 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (#‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓𝑘)))) = (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘)))))
2015, 19breq12d 4626 . . . . . . . 8 (((𝑔 = 𝐺𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑠 ≤ (#‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓𝑘)))) ↔ 𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘))))))
2120ralbidv 2980 . . . . . . 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 746 . . . . . 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 3454 . . . . 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 2738 . . . 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 482 . . 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 3198 . . . 4 (𝐺𝑊𝐺 ∈ V)
2726adantr 481 . . 3 ((𝐺𝑊𝑆 ∈ ℕ0*) → 𝐺 ∈ V)
28 simpr 477 . . 3 ((𝐺𝑊𝑆 ∈ ℕ0*) → 𝑆 ∈ ℕ0*)
29 df-rab 2916 . . . 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 6158 . . . . . . . 8 (iEdg‘𝐺) ∈ V
3130dmex 7046 . . . . . . 7 dom (iEdg‘𝐺) ∈ V
3231wrdexi 13256 . . . . . 6 Word dom (iEdg‘𝐺) ∈ V
3332rabex 4773 . . . . 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 2703 . . 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 6741 . 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 2630 . . . . . . . 8 (iEdg‘𝐺) = 𝐼
3938a1i 11 . . . . . . 7 ((𝐺𝑊𝑆 ∈ ℕ0*) → (iEdg‘𝐺) = 𝐼)
4039dmeqd 5286 . . . . . 6 ((𝐺𝑊𝑆 ∈ ℕ0*) → dom (iEdg‘𝐺) = dom 𝐼)
41 wrdeq 13266 . . . . . 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 2684 . . . 4 ((𝐺𝑊𝑆 ∈ ℕ0*) → (𝑓 ∈ Word dom (iEdg‘𝐺) ↔ 𝑓 ∈ Word dom 𝐼))
4439fveq1d 6150 . . . . . . . 8 ((𝐺𝑊𝑆 ∈ ℕ0*) → ((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) = (𝐼‘(𝑓‘(𝑘 − 1))))
4539fveq1d 6150 . . . . . . . 8 ((𝐺𝑊𝑆 ∈ ℕ0*) → ((iEdg‘𝐺)‘(𝑓𝑘)) = (𝐼‘(𝑓𝑘)))
4644, 45ineq12d 3793 . . . . . . 7 ((𝐺𝑊𝑆 ∈ ℕ0*) → (((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘))) = ((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓𝑘))))
4746fveq2d 6152 . . . . . 6 ((𝐺𝑊𝑆 ∈ ℕ0*) → (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘)))) = (#‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓𝑘)))))
4847breq2d 4625 . . . . 5 ((𝐺𝑊𝑆 ∈ ℕ0*) → (𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓𝑘)))) ↔ 𝑆 ≤ (#‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓𝑘))))))
4948ralbidv 2980 . . . 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 746 . . 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 2738 . 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 2655 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 384   = wceq 1480  wcel 1987  {cab 2607  wral 2907  {crab 2911  Vcvv 3186  [wsbc 3417  cin 3554   class class class wbr 4613  dom cdm 5074  cfv 5847  (class class class)co 6604  cmpt2 6606  1c1 9881  cle 10019  cmin 10210  0*cxnn0 11307  ..^cfzo 12406  #chash 13057  Word cword 13230  iEdgciedg 25775   EdgWalks cewlks 26361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-ewlks 26364
This theorem is referenced by:  isewlk  26368
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