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Theorem rusgrnumwwlklem 28915
Description: Lemma for rusgrnumwwlk 28920 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 ↩ (♯‘{𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑣}))
Assertion
Ref Expression
rusgrnumwwlklem ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (♯‘{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑃}))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑛,𝑁,𝑣,𝑀   𝑃,𝑛,𝑣,𝑀   𝑛,𝑉,𝑣,𝑀
Allowed substitution hints:   𝐿(𝑀,𝑣,𝑛)

Proof of Theorem rusgrnumwwlklem
StepHypRef Expression
1 oveq1 7364 . . . . 5 (𝑛 = 𝑁 → (𝑛 WWalksN 𝐺) = (𝑁 WWalksN 𝐺))
21adantl 482 . . . 4 ((𝑣 = 𝑃 ∧ 𝑛 = 𝑁) → (𝑛 WWalksN 𝐺) = (𝑁 WWalksN 𝐺))
3 eqeq2 2748 . . . . 5 (𝑣 = 𝑃 → ((𝑀‘0) = 𝑣 ↔ (𝑀‘0) = 𝑃))
43adantr 481 . . . 4 ((𝑣 = 𝑃 ∧ 𝑛 = 𝑁) → ((𝑀‘0) = 𝑣 ↔ (𝑀‘0) = 𝑃))
52, 4rabeqbidv 3424 . . 3 ((𝑣 = 𝑃 ∧ 𝑛 = 𝑁) → {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑣} = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑃})
65fveq2d 6846 . 2 ((𝑣 = 𝑃 ∧ 𝑛 = 𝑁) → (♯‘{𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑣}) = (♯‘{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑃}))
7 rusgrnumwwlk.l . 2 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↩ (♯‘{𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑣}))
8 fvex 6855 . 2 (♯‘{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑃}) ∈ V
96, 7, 8ovmpoa 7510 1 ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (♯‘{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑃}))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3407  â€˜cfv 6496  (class class class)co 7357   ∈ cmpo 7359  0cc0 11051  â„•0cn0 12413  â™¯chash 14230  Vtxcvtx 27947   WWalksN cwwlksn 28771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362
This theorem is referenced by:  rusgrnumwwlkb0  28916  rusgrnumwwlkb1  28917  rusgr0edg  28918  rusgrnumwwlks  28919  rusgrnumwwlkg  28921
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