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Theorem rusgranumwlklem1 26476
 Description: Lemma 1 for rusgranumwlk 26484. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
Hypothesis
Ref Expression
rusgranumwlk.w 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})
Assertion
Ref Expression
rusgranumwlklem1 (𝑅 ∈ (𝑊𝑁) → (𝑅 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑅)) = 𝑁))
Distinct variable groups:   𝐸,𝑐,𝑛   𝑁,𝑐,𝑛   𝑅,𝑐   𝑉,𝑐,𝑛
Allowed substitution hints:   𝑅(𝑛)   𝑊(𝑛,𝑐)

Proof of Theorem rusgranumwlklem1
StepHypRef Expression
1 rusgranumwlk.w . . 3 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})
2 ovex 6577 . . . 4 (𝑉 Walks 𝐸) ∈ V
32a1i 11 . . 3 (𝑁 ∈ ℕ0 → (𝑉 Walks 𝐸) ∈ V)
41, 3elfvmptrab 6214 . 2 (𝑅 ∈ (𝑊𝑁) → (𝑁 ∈ ℕ0𝑅 ∈ (𝑉 Walks 𝐸)))
52rabex 4740 . . . . . . 7 {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁} ∈ V
65a1i 11 . . . . . 6 (𝑅 ∈ (𝑉 Walks 𝐸) → {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁} ∈ V)
7 eqeq2 2621 . . . . . . . 8 (𝑛 = 𝑁 → ((#‘(1st𝑐)) = 𝑛 ↔ (#‘(1st𝑐)) = 𝑁))
87rabbidv 3164 . . . . . . 7 (𝑛 = 𝑁 → {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛} = {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁})
98, 1fvmptg 6189 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁} ∈ V) → (𝑊𝑁) = {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁})
106, 9sylan2 490 . . . . 5 ((𝑁 ∈ ℕ0𝑅 ∈ (𝑉 Walks 𝐸)) → (𝑊𝑁) = {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁})
1110eleq2d 2673 . . . 4 ((𝑁 ∈ ℕ0𝑅 ∈ (𝑉 Walks 𝐸)) → (𝑅 ∈ (𝑊𝑁) ↔ 𝑅 ∈ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁}))
12 fveq2 6103 . . . . . . . 8 (𝑐 = 𝑅 → (1st𝑐) = (1st𝑅))
1312fveq2d 6107 . . . . . . 7 (𝑐 = 𝑅 → (#‘(1st𝑐)) = (#‘(1st𝑅)))
1413eqeq1d 2612 . . . . . 6 (𝑐 = 𝑅 → ((#‘(1st𝑐)) = 𝑁 ↔ (#‘(1st𝑅)) = 𝑁))
1514elrab 3331 . . . . 5 (𝑅 ∈ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁} ↔ (𝑅 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑅)) = 𝑁))
16 simpr 476 . . . . . 6 ((𝑅 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑅)) = 𝑁) → (#‘(1st𝑅)) = 𝑁)
1716a1i 11 . . . . 5 ((𝑁 ∈ ℕ0𝑅 ∈ (𝑉 Walks 𝐸)) → ((𝑅 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑅)) = 𝑁) → (#‘(1st𝑅)) = 𝑁))
1815, 17syl5bi 231 . . . 4 ((𝑁 ∈ ℕ0𝑅 ∈ (𝑉 Walks 𝐸)) → (𝑅 ∈ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁} → (#‘(1st𝑅)) = 𝑁))
1911, 18sylbid 229 . . 3 ((𝑁 ∈ ℕ0𝑅 ∈ (𝑉 Walks 𝐸)) → (𝑅 ∈ (𝑊𝑁) → (#‘(1st𝑅)) = 𝑁))
20 simpr 476 . . 3 ((𝑁 ∈ ℕ0𝑅 ∈ (𝑉 Walks 𝐸)) → 𝑅 ∈ (𝑉 Walks 𝐸))
2119, 20jctild 564 . 2 ((𝑁 ∈ ℕ0𝑅 ∈ (𝑉 Walks 𝐸)) → (𝑅 ∈ (𝑊𝑁) → (𝑅 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑅)) = 𝑁)))
224, 21mpcom 37 1 (𝑅 ∈ (𝑊𝑁) → (𝑅 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑅)) = 𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  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-8 1979  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-pow 4769  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-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  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-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552 This theorem is referenced by: (None)
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