Theorem rusgrnumwwlklem 30442

Description: Lemma for rusgrnumwwlk 30447 etc. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 7-May-2021.)

Hypotheses

Ref	Expression
rusgrnumwwlk.v	⊢ 𝑉 = (Vtx‘𝐺)
rusgrnumwwlk.l	⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))

Assertion

Ref	Expression
rusgrnumwwlklem	⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))

Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑁,𝑣,𝑤   𝑃,𝑛,𝑣,𝑤   𝑛,𝑉,𝑣,𝑤

Allowed substitution hints:   𝐿(𝑤, 𝑣, 𝑛)

Proof of Theorem rusgrnumwwlklem

Step	Hyp	Ref	Expression
1		oveq1 7421	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 WWalksN 𝐺) = (𝑁 WWalksN 𝐺))
2	1	adantl 487	. . . 4 ⊢ ((𝑣 = 𝑃 ∧ 𝑛 = 𝑁) → (𝑛 WWalksN 𝐺) = (𝑁 WWalksN 𝐺))
3		eqeq2 2772	. . . . 5 ⊢ (𝑣 = 𝑃 → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑃))
4	3	adantr 486	. . . 4 ⊢ ((𝑣 = 𝑃 ∧ 𝑛 = 𝑁) → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑃))
5	2, 4	rabeqbidv 3429	. . 3 ⊢ ((𝑣 = 𝑃 ∧ 𝑛 = 𝑁) → {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃})
6	5	fveq2d 6883	. 2 ⊢ ((𝑣 = 𝑃 ∧ 𝑛 = 𝑁) → (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
7		rusgrnumwwlk.l	. 2 ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
8		fvex 6892	. 2 ⊢ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) ∈ V
9	6, 7, 8	ovmpoa 7569	1 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3412  ‘cfv 6533  (class class class)co 7414   ∈ cmpo 7416  0cc0 11125  ℕ₀cn0 12529  ♯chash 14395  Vtxcvtx 29454   WWalksN cwwlksn 30295

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 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419

This theorem is used by:  rusgrnumwwlkb0  30443  rusgrnumwwlkb1  30444  rusgr0edg  30445  rusgrnumwwlks  30446  rusgrnumwwlkg  30448