Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  av-numclwwlkovf2 Structured version   Visualization version   GIF version

Theorem av-numclwwlkovf2 41515
 Description: Value of operation 𝐹 for argument 2. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 28-May-2021.)
Hypotheses
Ref Expression
av-numclwwlkovf.f 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑣})
av-numclwwlkffin.v 𝑉 = (Vtx‘𝐺)
av-numclwwlkovfel2.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
av-numclwwlkovf2 ((𝐺 ∈ USGraph ∧ 𝑋𝑉) → (𝑋𝐹2) = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)})
Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑤   𝑤,𝑉
Allowed substitution hints:   𝐸(𝑤,𝑣,𝑛)   𝐹(𝑤,𝑣,𝑛)

Proof of Theorem av-numclwwlkovf2
StepHypRef Expression
1 simpr 476 . . 3 ((𝐺 ∈ USGraph ∧ 𝑋𝑉) → 𝑋𝑉)
2 2nn 11062 . . 3 2 ∈ ℕ
3 av-numclwwlkovf.f . . . 4 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑣})
43av-numclwwlkovf 41511 . . 3 ((𝑋𝑉 ∧ 2 ∈ ℕ) → (𝑋𝐹2) = {𝑤 ∈ (2 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋})
51, 2, 4sylancl 693 . 2 ((𝐺 ∈ USGraph ∧ 𝑋𝑉) → (𝑋𝐹2) = {𝑤 ∈ (2 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋})
6 clwwlksn2 41217 . . . . . 6 (𝑤 ∈ (2 ClWWalkSN 𝐺) ↔ ((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
76anbi1i 727 . . . . 5 ((𝑤 ∈ (2 ClWWalkSN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋))
87a1i 11 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑋𝑉) → ((𝑤 ∈ (2 ClWWalkSN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)))
9 anass 679 . . . . 5 (((𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸)) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ (((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸) ∧ (𝑤‘0) = 𝑋)))
10 df-3an 1033 . . . . . . . 8 (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ↔ (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺)) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
11 ancom 465 . . . . . . . . . 10 (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺)) ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 2))
12 av-numclwwlkffin.v . . . . . . . . . . . . . 14 𝑉 = (Vtx‘𝐺)
1312eqcomi 2619 . . . . . . . . . . . . 13 (Vtx‘𝐺) = 𝑉
1413wrdeqi 13183 . . . . . . . . . . . 12 Word (Vtx‘𝐺) = Word 𝑉
1514eleq2i 2680 . . . . . . . . . . 11 (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word 𝑉)
1615anbi1i 727 . . . . . . . . . 10 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 2) ↔ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = 2))
1711, 16bitri 263 . . . . . . . . 9 (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺)) ↔ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = 2))
18 av-numclwwlkovfel2.e . . . . . . . . . . 11 𝐸 = (Edg‘𝐺)
1918eqcomi 2619 . . . . . . . . . 10 (Edg‘𝐺) = 𝐸
2019eleq2i 2680 . . . . . . . . 9 ({(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺) ↔ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸)
2117, 20anbi12i 729 . . . . . . . 8 ((((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺)) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ↔ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = 2) ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸))
2210, 21bitri 263 . . . . . . 7 (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ↔ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = 2) ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸))
23 anass 679 . . . . . . 7 (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = 2) ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸)))
2422, 23bitri 263 . . . . . 6 (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸)))
2524anbi1i 727 . . . . 5 ((((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) ↔ ((𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸)) ∧ (𝑤‘0) = 𝑋))
26 df-3an 1033 . . . . . 6 (((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋) ↔ (((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸) ∧ (𝑤‘0) = 𝑋))
2726anbi2i 726 . . . . 5 ((𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)) ↔ (𝑤 ∈ Word 𝑉 ∧ (((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸) ∧ (𝑤‘0) = 𝑋)))
289, 25, 273bitr4i 291 . . . 4 ((((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)))
298, 28syl6bb 275 . . 3 ((𝐺 ∈ USGraph ∧ 𝑋𝑉) → ((𝑤 ∈ (2 ClWWalkSN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋))))
3029rabbidva2 3162 . 2 ((𝐺 ∈ USGraph ∧ 𝑋𝑉) → {𝑤 ∈ (2 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)})
315, 30eqtrd 2644 1 ((𝐺 ∈ USGraph ∧ 𝑋𝑉) → (𝑋𝐹2) = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  {cpr 4127  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  1c1 9816  ℕcn 10897  2c2 10947  #chash 12979  Word cword 13146  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379   ClWWalkSN cclwwlksn 41184 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-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-lsw 13155  df-clwwlks 41185  df-clwwlksn 41186 This theorem is referenced by:  av-numclwwlkovf2num  41516
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