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Theorem numclwwlkovf2 27107
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
numclwwlkovf.f 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
numclwwlkffin.v 𝑉 = (Vtx‘𝐺)
numclwwlkovfel2.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
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 numclwwlkovf2
StepHypRef Expression
1 simpr 477 . . 3 ((𝐺 ∈ USGraph ∧ 𝑋𝑉) → 𝑋𝑉)
2 2nn 11145 . . 3 2 ∈ ℕ
3 numclwwlkovf.f . . . 4 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
43numclwwlkovf 27103 . . 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 26810 . . . . . 6 (𝑤 ∈ (2 ClWWalksN 𝐺) ↔ ((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
76anbi1i 730 . . . . 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 680 . . . . 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 1038 . . . . . . . 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 466 . . . . . . . . . 10 (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺)) ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 2))
12 numclwwlkffin.v . . . . . . . . . . . . . 14 𝑉 = (Vtx‘𝐺)
1312eqcomi 2630 . . . . . . . . . . . . 13 (Vtx‘𝐺) = 𝑉
1413wrdeqi 13283 . . . . . . . . . . . 12 Word (Vtx‘𝐺) = Word 𝑉
1514eleq2i 2690 . . . . . . . . . . 11 (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word 𝑉)
1615anbi1i 730 . . . . . . . . . 10 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 2) ↔ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = 2))
1711, 16bitri 264 . . . . . . . . 9 (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺)) ↔ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = 2))
18 numclwwlkovfel2.e . . . . . . . . . . 11 𝐸 = (Edg‘𝐺)
1918eqcomi 2630 . . . . . . . . . 10 (Edg‘𝐺) = 𝐸
2019eleq2i 2690 . . . . . . . . 9 ({(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺) ↔ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸)
2117, 20anbi12i 732 . . . . . . . 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 264 . . . . . . 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 680 . . . . . . 7 (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = 2) ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸)))
2422, 23bitri 264 . . . . . 6 (((#‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸)))
2524anbi1i 730 . . . . 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 1038 . . . . . 6 (((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋) ↔ (((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸) ∧ (𝑤‘0) = 𝑋))
2726anbi2i 729 . . . . 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 292 . . . 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 276 . . 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 3178 . 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 2655 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {crab 2912  {cpr 4157  cfv 5857  (class class class)co 6615  cmpt2 6617  0cc0 9896  1c1 9897  cn 10980  2c2 11030  #chash 13073  Word cword 13246  Vtxcvtx 25808  Edgcedg 25873   USGraph cusgr 25971   ClWWalksN cclwwlksn 26777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423  df-hash 13074  df-word 13254  df-lsw 13255  df-clwwlks 26778  df-clwwlksn 26779
This theorem is referenced by:  numclwwlkovf2num  27108
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