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Theorem ewlksfval 27871
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 27868 . . . 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 6771 . . . . . 6 ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (iEdg‘𝑔) ∈ V)
4 simpr 484 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑖 = (iEdg‘𝑔))
5 fveq2 6756 . . . . . . . . . . . . 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 2778 . . . . . . . . . 10 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑖 = (iEdg‘𝐺))
98dmeqd 5803 . . . . . . . . 9 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → dom 𝑖 = dom (iEdg‘𝐺))
10 wrdeq 14167 . . . . . . . . 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 2824 . . . . . . 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 6758 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑖‘(𝑓‘(𝑘 − 1))) = ((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))))
168fveq1d 6758 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑖‘(𝑓‘𝑘)) = ((iEdg‘𝐺)‘(𝑓‘𝑘)))
1715, 16ineq12d 4144 . . . . . . . . . 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 6760 . . . . . . . . 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 5083 . . . . . . . 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 3120 . . . . . . 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 630 . . . . . 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 3756 . . . . 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 2808 . . . 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 3440 . . . 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 3072 . . . 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 6769 . . . . . . . 8 (iEdg‘𝐺) ∈ V
3029dmex 7732 . . . . . . 7 dom (iEdg‘𝐺) ∈ V
3130wrdexi 14157 . . . . . 6 Word dom (iEdg‘𝐺) ∈ V
3231rabex 5251 . . . . 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 2844 . . 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 7403 . 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 2747 . . . . . . . 8 (iEdg‘𝐺) = 𝐼
3837a1i 11 . . . . . . 7 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (iEdg‘𝐺) = 𝐼)
3938dmeqd 5803 . . . . . 6 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → dom (iEdg‘𝐺) = dom 𝐼)
40 wrdeq 14167 . . . . . 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 2824 . . . 4 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (𝑓 ∈ Word dom (iEdg‘𝐺) ↔ 𝑓 ∈ Word dom 𝐼))
4338fveq1d 6758 . . . . . . . 8 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → ((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) = (𝐼‘(𝑓‘(𝑘 − 1))))
4438fveq1d 6758 . . . . . . . 8 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → ((iEdg‘𝐺)‘(𝑓‘𝑘)) = (𝐼‘(𝑓‘𝑘)))
4543, 44ineq12d 4144 . . . . . . 7 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘))) = ((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘))))
4645fveq2d 6760 . . . . . 6 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))) = (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘)))))
4746breq2d 5082 . . . . 5 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))) ↔ 𝑆 ≤ (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘))))))
4847ralbidv 3120 . . . 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 630 . . 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 2808 . 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 2778 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 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  {crab 3067  Vcvv 3422  [wsbc 3711   ∩ cin 3882   class class class wbr 5070  dom cdm 5580  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  1c1 10803   ≤ cle 10941   − cmin 11135  ℕ0*cxnn0 12235  ..^cfzo 13311  ♯chash 13972  Word cword 14145  iEdgciedg 27270   EdgWalks cewlks 27865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-hash 13973  df-word 14146  df-ewlks 27868
This theorem is referenced by:  isewlk  27872
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