MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ewlksfval Structured version   Visualization version   GIF version
Theorem ewlksfval 28896
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 28893 . . . 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 6906 . . . . . 6 ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (iEdg‘𝑔) ∈ V)
4 simpr 485 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑖 = (iEdg‘𝑔))
5 fveq2 6891 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
65adantr 481 . . . . . . . . . . . 12 ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (iEdg‘𝑔) = (iEdg‘𝐺))
76adantr 481 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (iEdg‘𝑔) = (iEdg‘𝐺))
84, 7eqtrd 2772 . . . . . . . . . 10 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑖 = (iEdg‘𝐺))
98dmeqd 5905 . . . . . . . . 9 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → dom 𝑖 = dom (iEdg‘𝐺))
10 wrdeq 14488 . . . . . . . . 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 2819 . . . . . . 7 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑓 ∈ Word dom 𝑖 ↔ 𝑓 ∈ Word dom (iEdg‘𝐺)))
13 simpr 485 . . . . . . . . . 10 ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
1413adantr 481 . . . . . . . . 9 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑠 = 𝑆)
158fveq1d 6893 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑖‘(𝑓‘(𝑘 − 1))) = ((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))))
168fveq1d 6893 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑖‘(𝑓‘𝑘)) = ((iEdg‘𝐺)‘(𝑓‘𝑘)))
1715, 16ineq12d 4213 . . . . . . . . . 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 6895 . . . . . . . . 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 5161 . . . . . . . 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 3177 . . . . . . 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 631 . . . . . 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 3822 . . . . 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 2801 . . . 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 482 . . 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 3492 . . . 4 (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
2625adantr 481 . . 3 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → 𝐺 ∈ V)
27 simpr 485 . . 3 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → 𝑆 ∈ ℕ0*)
28 df-rab 3433 . . . 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 6904 . . . . . . . 8 (iEdg‘𝐺) ∈ V
3029dmex 7904 . . . . . . 7 dom (iEdg‘𝐺) ∈ V
3130wrdexi 14478 . . . . . 6 Word dom (iEdg‘𝐺) ∈ V
3231rabex 5332 . . . . 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 2838 . . 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 7562 . 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 2741 . . . . . . . 8 (iEdg‘𝐺) = 𝐼
3837a1i 11 . . . . . . 7 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (iEdg‘𝐺) = 𝐼)
3938dmeqd 5905 . . . . . 6 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → dom (iEdg‘𝐺) = dom 𝐼)
40 wrdeq 14488 . . . . . 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 2819 . . . 4 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (𝑓 ∈ Word dom (iEdg‘𝐺) ↔ 𝑓 ∈ Word dom 𝐼))
4338fveq1d 6893 . . . . . . . 8 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → ((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) = (𝐼‘(𝑓‘(𝑘 − 1))))
4438fveq1d 6893 . . . . . . . 8 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → ((iEdg‘𝐺)‘(𝑓‘𝑘)) = (𝐼‘(𝑓‘𝑘)))
4543, 44ineq12d 4213 . . . . . . 7 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘))) = ((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘))))
4645fveq2d 6895 . . . . . 6 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))) = (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘)))))
4746breq2d 5160 . . . . 5 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))) ↔ 𝑆 ≤ (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘))))))
4847ralbidv 3177 . . . 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 631 . . 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 2801 . 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 2772 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 396   = wceq 1541   ∈ wcel 2106  {cab 2709  ∀wral 3061  {crab 3432  Vcvv 3474  [wsbc 3777   ∩ cin 3947   class class class wbr 5148  dom cdm 5676  ‘cfv 6543  (class class class)co 7411   ∈ cmpo 7413  1c1 11113   ≤ cle 11251   − cmin 11446  ℕ0*cxnn0 12546  ..^cfzo 13629  ♯chash 14292  Word cword 14466  iEdgciedg 28295   EdgWalks cewlks 28890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-n0 12475  df-z 12561  df-uz 12825  df-fz 13487  df-fzo 13630  df-hash 14293  df-word 14467  df-ewlks 28893
This theorem is referenced by:  isewlk  28897
  Copyright terms: Public domain W3C validator