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Theorem ewlksfval 30182
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 30179 . . . 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 6900 . . . . . 6 ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (iEdg‘𝑔) ∈ V)
4 simpr 490 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑖 = (iEdg‘𝑔))
5 fveq2 6885 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
65adantr 486 . . . . . . . . . . . 12 ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → (iEdg‘𝑔) = (iEdg‘𝐺))
76adantr 486 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (iEdg‘𝑔) = (iEdg‘𝐺))
84, 7eqtrd 2796 . . . . . . . . . 10 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑖 = (iEdg‘𝐺))
98dmeqd 5887 . . . . . . . . 9 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → dom 𝑖 = dom (iEdg‘𝐺))
10 wrdeq 14681 . . . . . . . . 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 2847 . . . . . . 7 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑓 ∈ Word dom 𝑖 ↔ 𝑓 ∈ Word dom (iEdg‘𝐺)))
13 simpr 490 . . . . . . . . . 10 ((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
1413adantr 486 . . . . . . . . 9 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → 𝑠 = 𝑆)
158fveq1d 6887 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑖‘(𝑓‘(𝑘 − 1))) = ((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))))
168fveq1d 6887 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑠 = 𝑆) ∧ 𝑖 = (iEdg‘𝑔)) → (𝑖‘(𝑓‘𝑘)) = ((iEdg‘𝐺)‘(𝑓‘𝑘)))
1715, 16ineq12d 4167 . . . . . . . . . 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 6889 . . . . . . . . 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 5116 . . . . . . . 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 3186 . . . . . . 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 644 . . . . . 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 3782 . . . . 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 2827 . . . 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 487 . . 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 3472 . . . 4 (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
2625adantr 486 . . 3 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → 𝐺 ∈ V)
27 simpr 490 . . 3 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → 𝑆 ∈ ℕ0*)
28 df-rab 3414 . . . 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 6898 . . . . . . . 8 (iEdg‘𝐺) ∈ V
3029dmex 7921 . . . . . . 7 dom (iEdg‘𝐺) ∈ V
3130wrdexi 14671 . . . . . 6 Word dom (iEdg‘𝐺) ∈ V
3231rabex 5300 . . . . 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 2866 . . 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 7572 . 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 2770 . . . . . . . 8 (iEdg‘𝐺) = 𝐼
3837a1i 11 . . . . . . 7 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (iEdg‘𝐺) = 𝐼)
3938dmeqd 5887 . . . . . 6 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → dom (iEdg‘𝐺) = dom 𝐼)
40 wrdeq 14681 . . . . . 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 2847 . . . 4 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (𝑓 ∈ Word dom (iEdg‘𝐺) ↔ 𝑓 ∈ Word dom 𝐼))
4338fveq1d 6887 . . . . . . . 8 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → ((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) = (𝐼‘(𝑓‘(𝑘 − 1))))
4438fveq1d 6887 . . . . . . . 8 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → ((iEdg‘𝐺)‘(𝑓‘𝑘)) = (𝐼‘(𝑓‘𝑘)))
4543, 44ineq12d 4167 . . . . . . 7 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘))) = ((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘))))
4645fveq2d 6889 . . . . . 6 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))) = (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘)))))
4746breq2d 5115 . . . . 5 ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝑓‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝑓‘𝑘)))) ↔ 𝑆 ≤ (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘))))))
4847ralbidv 3186 . . . 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 644 . . 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 2827 . 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 2796 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
This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  {crab 3413  Vcvv 3451  [wsbc 3739   ∩ cin 3898   class class class wbr 5103  dom cdm 5651  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1c1 11201   ≤ cle 11344   − cmin 11541  ℕ0*cxnn0 12679  ..^cfzo 13788  ♯chash 14474  Word cword 14658  iEdgciedg 29575   EdgWalks cewlks 30176
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-n0 12607  df-z 12694  df-uz 12966  df-fz 13640  df-fzo 13789  df-hash 14475  df-word 14659  df-ewlks 30179
This theorem is used by:  isewlk  30183
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