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Theorem rusgranumwlklem4 26479
 Description: Lemma 4 for rusgranumwlk 26484. (Contributed by Alexander van der Vekens, 24-Jul-2018.) (Proof shortened by AV, 5-May-2021.)
Hypotheses
Ref Expression
rusgranumwlk.w 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})
rusgranumwlk.l 𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))
Assertion
Ref Expression
rusgranumwlklem4 ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (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   𝑣,𝐸,𝑤
Allowed substitution hints:   𝑃(𝑐)   𝐿(𝑤,𝑣,𝑛,𝑐)   𝑊(𝑐)

Proof of Theorem rusgranumwlklem4
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 rusgranumwlk.w . . . 4 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})
2 rusgranumwlk.l . . . 4 𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))
31, 2rusgranumwlklem3 26478 . . 3 ((𝑃𝑉𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}))
433adant1 1072 . 2 ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}))
5 fveq2 6103 . . . . . . . 8 (𝑤 = 𝑣 → (1st𝑤) = (1st𝑣))
65fveq2d 6107 . . . . . . 7 (𝑤 = 𝑣 → (#‘(1st𝑤)) = (#‘(1st𝑣)))
76eqeq1d 2612 . . . . . 6 (𝑤 = 𝑣 → ((#‘(1st𝑤)) = 𝑁 ↔ (#‘(1st𝑣)) = 𝑁))
8 fveq2 6103 . . . . . . . 8 (𝑤 = 𝑣 → (2nd𝑤) = (2nd𝑣))
98fveq1d 6105 . . . . . . 7 (𝑤 = 𝑣 → ((2nd𝑤)‘0) = ((2nd𝑣)‘0))
109eqeq1d 2612 . . . . . 6 (𝑤 = 𝑣 → (((2nd𝑤)‘0) = 𝑃 ↔ ((2nd𝑣)‘0) = 𝑃))
117, 10anbi12d 743 . . . . 5 (𝑤 = 𝑣 → (((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃) ↔ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)))
1211cbvrabv 3172 . . . 4 {𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)} = {𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)}
1312a1i 11 . . 3 ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → {𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)} = {𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)})
1413fveq2d 6107 . 2 ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}) = (#‘{𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)}))
15 ovex 6577 . . . 4 (𝑉 Walks 𝐸) ∈ V
1615rabex 4740 . . 3 {𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)} ∈ V
17 wlkiswwlkbij2 26249 . . 3 ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃})
18 hasheqf1oi 13002 . . 3 ({𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)} ∈ V → (∃𝑓 𝑓:{𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} → (#‘{𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)}) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃})))
1916, 17, 18mpsyl 66 . 2 ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → (#‘{𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)}) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}))
204, 14, 193eqtrd 2648 1 ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {crab 2900  Vcvv 3173   class class class wbr 4583   ↦ cmpt 4643  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  0cc0 9815  ℕ0cn0 11169  #chash 12979   USGrph cusg 25859   Walks cwalk 26026   WWalksN cwwlkn 26206 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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  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-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  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-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-wwlk 26207  df-wwlkn 26208 This theorem is referenced by:  rusgranumwlkb0  26480  rusgranumwlkb1  26481  rusgra0edg  26482  rusgranumwlks  26483
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