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) ∣ (#‘(1st ‘𝑐)) = 𝑛})
rusgranumwlk.l	⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊‘𝑛) ∣ ((2nd ‘𝑤)‘0) = 𝑣}))

Assertion

Ref	Expression
rusgranumwlklem3	⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘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

Step	Hyp	Ref	Expression
1		rusgranumwlk.w	. . 3 ⊢ 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑛})
2		rusgranumwlk.l	. . 3 ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊‘𝑛) ∣ ((2nd ‘𝑤)‘0) = 𝑣}))
3	1, 2	rusgranumwlklem2 26477	. 2 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑊‘𝑁) ∣ ((2nd ‘𝑤)‘0) = 𝑃}))
4		eqeq2 2621	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → ((#‘(1st ‘𝑐)) = 𝑛 ↔ (#‘(1st ‘𝑐)) = 𝑁))
5	4	rabbidv 3164	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑛} = {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑁})
6		ovex 6577	. . . . . . . . . . 11 ⊢ (𝑉 Walks 𝐸) ∈ V
7	6	rabex 4740	. . . . . . . . . 10 ⊢ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑁} ∈ V
8	5, 1, 7	fvmpt 6191	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑊‘𝑁) = {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑁})
9	8	eleq2d 2673	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑤 ∈ (𝑊‘𝑁) ↔ 𝑤 ∈ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑁}))
10		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑤 → (1st ‘𝑐) = (1st ‘𝑤))
11	10	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑐 = 𝑤 → (#‘(1st ‘𝑐)) = (#‘(1st ‘𝑤)))
12	11	eqeq1d 2612	. . . . . . . . 9 ⊢ (𝑐 = 𝑤 → ((#‘(1st ‘𝑐)) = 𝑁 ↔ (#‘(1st ‘𝑤)) = 𝑁))
13	12	elrab 3331	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑁} ↔ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘𝑤)) = 𝑁))
14	9, 13	syl6bb 275	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑤 ∈ (𝑊‘𝑁) ↔ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘𝑤)) = 𝑁)))
15	14	adantl 481	. . . . . 6 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑤 ∈ (𝑊‘𝑁) ↔ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘𝑤)) = 𝑁)))
16	15	anbi1d 737	. . . . 5 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ((𝑤 ∈ (𝑊‘𝑁) ∧ ((2nd ‘𝑤)‘0) = 𝑃) ↔ ((𝑤 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘𝑤)) = 𝑁) ∧ ((2nd ‘𝑤)‘0) = 𝑃)))
17		anass 679	. . . . 5 ⊢ (((𝑤 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘𝑤)) = 𝑁) ∧ ((2nd ‘𝑤)‘0) = 𝑃) ↔ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)))
18	16, 17	syl6bb 275	. . . 4 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ((𝑤 ∈ (𝑊‘𝑁) ∧ ((2nd ‘𝑤)‘0) = 𝑃) ↔ (𝑤 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃))))
19	18	rabbidva2 3162	. . 3 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → {𝑤 ∈ (𝑊‘𝑁) ∣ ((2nd ‘𝑤)‘0) = 𝑃} = {𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)})
20	19	fveq2d 6107	. 2 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (#‘{𝑤 ∈ (𝑊‘𝑁) ∣ ((2nd ‘𝑤)‘0) = 𝑃}) = (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)}))
21	3, 20	eqtrd 2644	1 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)}))

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  1^st c1st 7057  2^nd 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