Theorem rusgrnumwwlklem 27749

Description: Lemma for rusgrnumwwlk 27754 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 7163	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 WWalksN 𝐺) = (𝑁 WWalksN 𝐺))
2	1	adantl 484	. . . 4 ⊢ ((𝑣 = 𝑃 ∧ 𝑛 = 𝑁) → (𝑛 WWalksN 𝐺) = (𝑁 WWalksN 𝐺))
3		eqeq2 2833	. . . . 5 ⊢ (𝑣 = 𝑃 → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑃))
4	3	adantr 483	. . . 4 ⊢ ((𝑣 = 𝑃 ∧ 𝑛 = 𝑁) → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑃))
5	2, 4	rabeqbidv 3485	. . 3 ⊢ ((𝑣 = 𝑃 ∧ 𝑛 = 𝑁) → {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃})
6	5	fveq2d 6674	. 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 6683	. 2 ⊢ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) ∈ V
9	6, 7, 8	ovmpoa 7305	1 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {crab 3142  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  0cc0 10537  ℕ₀cn0 11898  ♯chash 13691  Vtxcvtx 26781   WWalksN cwwlksn 27604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161

This theorem is referenced by:  rusgrnumwwlkb0  27750  rusgrnumwwlkb1  27751  rusgr0edg  27752  rusgrnumwwlks  27753  rusgrnumwwlkg  27755