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Theorem ewlksfval 29580
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 29577 . . . 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 6837 . . . . . 6 ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (iEdg‘𝑔) ∈ V)
4 simpr 484 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑖 = (iEdg‘𝑔))
5 fveq2 6822 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
65adantr 480 . . . . . . . . . . . 12 ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (iEdg‘𝑔) = (iEdg‘𝐺))
76adantr 480 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (iEdg‘𝑔) = (iEdg‘𝐺))
84, 7eqtrd 2766 . . . . . . . . . 10 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑖 = (iEdg‘𝐺))
98dmeqd 5844 . . . . . . . . 9 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → dom 𝑖 = dom (iEdg‘𝐺))
10 wrdeq 14443 . . . . . . . . 9 (dom 𝑖 = dom (iEdg‘𝐺) → Word dom 𝑖 = Word dom (iEdg‘𝐺))
119, 10syl 17 . . . . . . . 8 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → Word dom 𝑖 = Word dom (iEdg‘𝐺))
1211eleq2d 2817 . . . . . . 7 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑓 ∈ Word dom 𝑖 ↔ 𝑓 ∈ Word dom (iEdg‘𝐺)))
13 simpr 484 . . . . . . . . . 10 ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
1413adantr 480 . . . . . . . . 9 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑠 = 𝑆)
158fveq1d 6824 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑖‘(𝑓‘(𝑘 − 1))) = ((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))))
168fveq1d 6824 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑖‘(𝑓‘𝑘)) = ((iEdg‘𝐺)‘(𝑓‘𝑘)))
1715, 16ineq12d 4168 . . . . . . . . . 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 6826 . . . . . . . . 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 5102 . . . . . . . 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 3155 . . . . . . 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 632 . . . . . 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 3780 . . . . 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 2797 . . . 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 481 . . 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 3457 . . . 4 (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
2625adantr 480 . . 3 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → 𝐺 ∈ V)
27 simpr 484 . . 3 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → 𝑆 ∈ ℕ0*)
28 df-rab 3396 . . . 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 6835 . . . . . . . 8 (iEdg‘𝐺) ∈ V
3029dmex 7839 . . . . . . 7 dom (iEdg‘𝐺) ∈ V
3130wrdexi 14433 . . . . . 6 Word dom (iEdg‘𝐺) ∈ V
3231rabex 5275 . . . . 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 2836 . . 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 7498 . 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 2740 . . . . . . . 8 (iEdg‘𝐺) = 𝐼
3837a1i 11 . . . . . . 7 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (iEdg‘𝐺) = 𝐼)
3938dmeqd 5844 . . . . . 6 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → dom (iEdg‘𝐺) = dom 𝐼)
40 wrdeq 14443 . . . . . 6 (dom (iEdg‘𝐺) = dom 𝐼 → Word dom (iEdg‘𝐺) = Word dom 𝐼)
4139, 40syl 17 . . . . 5 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → Word dom (iEdg‘𝐺) = Word dom 𝐼)
4241eleq2d 2817 . . . 4 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (𝑓 ∈ Word dom (iEdg‘𝐺) ↔ 𝑓 ∈ Word dom 𝐼))
4338fveq1d 6824 . . . . . . . 8 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → ((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) = (𝐼‘(𝑓‘(𝑘 − 1))))
4438fveq1d 6824 . . . . . . . 8 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → ((iEdg‘𝐺)‘(𝑓‘𝑘)) = (𝐼‘(𝑓‘𝑘)))
4543, 44ineq12d 4168 . . . . . . 7 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘))) = ((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘))))
4645fveq2d 6826 . . . . . 6 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))) = (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘)))))
4746breq2d 5101 . . . . 5 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))) ↔ 𝑆 ≤ (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘))))))
4847ralbidv 3155 . . . 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 632 . . 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 2797 . 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 2766 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 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  {crab 3395  Vcvv 3436  [wsbc 3736   ∩ cin 3896   class class class wbr 5089  dom cdm 5614  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  1c1 11007   ≤ cle 11147   − cmin 11344  ℕ0*cxnn0 12454  ..^cfzo 13554  ♯chash 14237  Word cword 14420  iEdgciedg 28975   EdgWalks cewlks 29574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-n0 12382  df-z 12469  df-uz 12733  df-fz 13408  df-fzo 13555  df-hash 14238  df-word 14421  df-ewlks 29577
This theorem is referenced by:  isewlk  29581
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