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Theorem rusgranumwlklem3 26478
 Description: Lemma 3 for rusgranumwlk 26484. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
Hypotheses
Ref Expression
rusgranumwlk.w 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})
rusgranumwlk.l 𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))
Assertion
Ref Expression
rusgranumwlklem3 ((𝑃𝑉𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}))
Distinct variable groups:   𝐸,𝑐,𝑛   𝑁,𝑐,𝑛   𝑉,𝑐,𝑛   𝑣,𝑁,𝑤   𝑃,𝑛,𝑣,𝑤   𝑣,𝑉   𝑛,𝑊,𝑣,𝑤   𝑤,𝑉,𝑐
Allowed substitution hints:   𝑃(𝑐)   𝐸(𝑤,𝑣)   𝐿(𝑤,𝑣,𝑛,𝑐)   𝑊(𝑐)

Proof of Theorem rusgranumwlklem3
StepHypRef Expression
1 rusgranumwlk.w . . 3 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})
2 rusgranumwlk.l . . 3 𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))
31, 2rusgranumwlklem2 26477 . 2 ((𝑃𝑉𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑊𝑁) ∣ ((2nd𝑤)‘0) = 𝑃}))
4 eqeq2 2621 . . . . . . . . . . 11 (𝑛 = 𝑁 → ((#‘(1st𝑐)) = 𝑛 ↔ (#‘(1st𝑐)) = 𝑁))
54rabbidv 3164 . . . . . . . . . 10 (𝑛 = 𝑁 → {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛} = {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁})
6 ovex 6577 . . . . . . . . . . 11 (𝑉 Walks 𝐸) ∈ V
76rabex 4740 . . . . . . . . . 10 {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁} ∈ V
85, 1, 7fvmpt 6191 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑊𝑁) = {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁})
98eleq2d 2673 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑤 ∈ (𝑊𝑁) ↔ 𝑤 ∈ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁}))
10 fveq2 6103 . . . . . . . . . . 11 (𝑐 = 𝑤 → (1st𝑐) = (1st𝑤))
1110fveq2d 6107 . . . . . . . . . 10 (𝑐 = 𝑤 → (#‘(1st𝑐)) = (#‘(1st𝑤)))
1211eqeq1d 2612 . . . . . . . . 9 (𝑐 = 𝑤 → ((#‘(1st𝑐)) = 𝑁 ↔ (#‘(1st𝑤)) = 𝑁))
1312elrab 3331 . . . . . . . 8 (𝑤 ∈ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁} ↔ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑤)) = 𝑁))
149, 13syl6bb 275 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝑤 ∈ (𝑊𝑁) ↔ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑤)) = 𝑁)))
1514adantl 481 . . . . . 6 ((𝑃𝑉𝑁 ∈ ℕ0) → (𝑤 ∈ (𝑊𝑁) ↔ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑤)) = 𝑁)))
1615anbi1d 737 . . . . 5 ((𝑃𝑉𝑁 ∈ ℕ0) → ((𝑤 ∈ (𝑊𝑁) ∧ ((2nd𝑤)‘0) = 𝑃) ↔ ((𝑤 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑤)) = 𝑁) ∧ ((2nd𝑤)‘0) = 𝑃)))
17 anass 679 . . . . 5 (((𝑤 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑤)) = 𝑁) ∧ ((2nd𝑤)‘0) = 𝑃) ↔ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)))
1816, 17syl6bb 275 . . . 4 ((𝑃𝑉𝑁 ∈ ℕ0) → ((𝑤 ∈ (𝑊𝑁) ∧ ((2nd𝑤)‘0) = 𝑃) ↔ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃))))
1918rabbidva2 3162 . . 3 ((𝑃𝑉𝑁 ∈ ℕ0) → {𝑤 ∈ (𝑊𝑁) ∣ ((2nd𝑤)‘0) = 𝑃} = {𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)})
2019fveq2d 6107 . 2 ((𝑃𝑉𝑁 ∈ ℕ0) → (#‘{𝑤 ∈ (𝑊𝑁) ∣ ((2nd𝑤)‘0) = 𝑃}) = (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}))
213, 20eqtrd 2644 1 ((𝑃𝑉𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  0cc0 9815  ℕ0cn0 11169  #chash 12979   Walks cwalk 26026 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  rusgranumwlklem4  26479  rusgranumwlkg  26485
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