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Theorem ewlksfval 29892
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 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (𝐺 EdgWalks 𝑆) = {𝑓 ∣ (𝑓 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘)))))})
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 29889 . . . 4 EdgWalks = (𝑔 ∈ V, 𝑠 ∈ ℕ0* ↦ {𝑓 ∣ [(iEdg‘𝑔) / 𝑖](𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑠 ≤ (♯‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓‘𝑘)))))})
21a1i 11 . . 3 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → EdgWalks = (𝑔 ∈ V, 𝑠 ∈ ℕ0* ↦ {𝑓 ∣ [(iEdg‘𝑔) / 𝑖](𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑠 ≤ (♯‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓‘𝑘)))))}))
3 fvexd 6897 . . . . . 6 ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (iEdg‘𝑔) ∈ V)
4 simpr 489 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑖 = (iEdg‘𝑔))
5 fveq2 6882 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
65adantr 485 . . . . . . . . . . . 12 ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (iEdg‘𝑔) = (iEdg‘𝐺))
76adantr 485 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (iEdg‘𝑔) = (iEdg‘𝐺))
84, 7eqtrd 2804 . . . . . . . . . 10 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑖 = (iEdg‘𝐺))
98dmeqd 5896 . . . . . . . . 9 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → dom 𝑖 = dom (iEdg‘𝐺))
10 wrdeq 14573 . . . . . . . . 9 (dom 𝑖 = dom (iEdg‘𝐺) → Word dom 𝑖 = Word dom (iEdg‘𝐺))
119, 10syl 18 . . . . . . . 8 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → Word dom 𝑖 = Word dom (iEdg‘𝐺))
1211eleq2d 2855 . . . . . . 7 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑓 ∈ Word dom 𝑖 ↔ 𝑓 ∈ Word dom (iEdg‘𝐺)))
13 simpr 489 . . . . . . . . . 10 ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
1413adantr 485 . . . . . . . . 9 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑠 = 𝑆)
158fveq1d 6884 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑖‘(𝑓‘(𝑘 − 1))) = ((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))))
168fveq1d 6884 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑖‘(𝑓‘𝑘)) = ((iEdg‘𝐺)‘(𝑓‘𝑘)))
1715, 16ineq12d 4182 . . . . . . . . . 10 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → ((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓‘𝑘))) = (((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘))))
1817fveq2d 6886 . . . . . . . . 9 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (♯‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓‘𝑘)))) = (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))))
1914, 18breq12d 5126 . . . . . . . 8 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑠 ≤ (♯‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓‘𝑘)))) ↔ 𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘))))))
2019ralbidv 3194 . . . . . . 7 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (∀𝑘 ∈ (1..^(♯‘𝑓))𝑠 ≤ (♯‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓‘𝑘)))) ↔ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘))))))
2112, 20anbi12d 643 . . . . . 6 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → ((𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑠 ≤ (♯‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓‘𝑘))))) ↔ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))))))
223, 21sbcied 3796 . . . . 5 ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → ([(iEdg‘𝑔) / 𝑖](𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑠 ≤ (♯‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓‘𝑘))))) ↔ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))))))
2322abbidv 2835 . . . 4 ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → {𝑓 ∣ [(iEdg‘𝑔) / 𝑖](𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑠 ≤ (♯‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓‘𝑘)))))} = {𝑓 ∣ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))))})
2423adantl 486 . . 3 (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) ∧ (𝑔 = 𝐺 ∧ 𝑠 = 𝑆)) → {𝑓 ∣ [(iEdg‘𝑔) / 𝑖](𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑠 ≤ (♯‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓‘𝑘)))))} = {𝑓 ∣ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))))})
25 elex 3484 . . . 4 (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
2625adantr 485 . . 3 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → 𝐺 ∈ V)
27 simpr 489 . . 3 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → 𝑆 ∈ ℕ0*)
28 df-rab 3424 . . . 4 {𝑓 ∈ Word dom (iEdg‘𝐺) ∣ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘))))} = {𝑓 ∣ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))))}
29 fvex 6895 . . . . . . . 8 (iEdg‘𝐺) ∈ V
3029dmex 7906 . . . . . . 7 dom (iEdg‘𝐺) ∈ V
3130wrdexi 14563 . . . . . 6 Word dom (iEdg‘𝐺) ∈ V
3231rabex 5310 . . . . 5 {𝑓 ∈ Word dom (iEdg‘𝐺) ∣ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘))))} ∈ V
3332a1i 11 . . . 4 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → {𝑓 ∈ Word dom (iEdg‘𝐺) ∣ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘))))} ∈ V)
3428, 33eqeltrrid 2874 . . 3 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → {𝑓 ∣ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))))} ∈ V)
352, 24, 26, 27, 34ovmpod 7563 . 2 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (𝐺 EdgWalks 𝑆) = {𝑓 ∣ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))))})
36 ewlksfval.i . . . . . . . . 9 𝐼 = (iEdg‘𝐺)
3736eqcomi 2778 . . . . . . . 8 (iEdg‘𝐺) = 𝐼
3837a1i 11 . . . . . . 7 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (iEdg‘𝐺) = 𝐼)
3938dmeqd 5896 . . . . . 6 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → dom (iEdg‘𝐺) = dom 𝐼)
40 wrdeq 14573 . . . . . 6 (dom (iEdg‘𝐺) = dom 𝐼 → Word dom (iEdg‘𝐺) = Word dom 𝐼)
4139, 40syl 18 . . . . 5 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → Word dom (iEdg‘𝐺) = Word dom 𝐼)
4241eleq2d 2855 . . . 4 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (𝑓 ∈ Word dom (iEdg‘𝐺) ↔ 𝑓 ∈ Word dom 𝐼))
4338fveq1d 6884 . . . . . . . 8 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → ((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) = (𝐼‘(𝑓‘(𝑘 − 1))))
4438fveq1d 6884 . . . . . . . 8 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → ((iEdg‘𝐺)‘(𝑓‘𝑘)) = (𝐼‘(𝑓‘𝑘)))
4543, 44ineq12d 4182 . . . . . . 7 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘))) = ((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘))))
4645fveq2d 6886 . . . . . 6 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))) = (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘)))))
4746breq2d 5125 . . . . 5 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))) ↔ 𝑆 ≤ (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘))))))
4847ralbidv 3194 . . . 4 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))) ↔ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘))))))
4942, 48anbi12d 643 . . 3 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → ((𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘))))) ↔ (𝑓 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘)))))))
5049abbidv 2835 . 2 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → {𝑓 ∣ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))))} = {𝑓 ∣ (𝑓 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘)))))})
5135, 50eqtrd 2804 1 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (𝐺 EdgWalks 𝑆) = {𝑓 ∣ (𝑓 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘)))))})
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  {crab 3423  Vcvv 3463  [wsbc 3753   ∩ cin 3912   class class class wbr 5113  dom cdm 5662  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  1c1 11101   ≤ cle 11244   − cmin 11441  ℕ0*cxnn0 12577  ..^cfzo 13682  ♯chash 14366  Word cword 14550  iEdgciedg 29288   EdgWalks cewlks 29886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-n0 12505  df-z 12592  df-uz 12863  df-fz 13536  df-fzo 13683  df-hash 14367  df-word 14551  df-ewlks 29889
This theorem is referenced by:  isewlk  29893
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