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Theorem wlkson 26608
Description: The set of walks between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 30-Dec-2020.) (Revised by AV, 22-Mar-2021.)
Hypothesis
Ref Expression
wlkson.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
wlkson ((𝐴𝑉𝐵𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)})
Distinct variable groups:   𝐴,𝑓,𝑝   𝐵,𝑓,𝑝   𝑓,𝐺,𝑝   𝑓,𝑉,𝑝

Proof of Theorem wlkson
Dummy variables 𝑎 𝑏 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wlkson.v . . . . 5 𝑉 = (Vtx‘𝐺)
211vgrex 25927 . . . 4 (𝐴𝑉𝐺 ∈ V)
32adantr 480 . . 3 ((𝐴𝑉𝐵𝑉) → 𝐺 ∈ V)
4 simpl 472 . . . 4 ((𝐴𝑉𝐵𝑉) → 𝐴𝑉)
54, 1syl6eleq 2740 . . 3 ((𝐴𝑉𝐵𝑉) → 𝐴 ∈ (Vtx‘𝐺))
6 simpr 476 . . . 4 ((𝐴𝑉𝐵𝑉) → 𝐵𝑉)
76, 1syl6eleq 2740 . . 3 ((𝐴𝑉𝐵𝑉) → 𝐵 ∈ (Vtx‘𝐺))
8 wksv 26571 . . . 4 {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V
98a1i 11 . . 3 ((𝐴𝑉𝐵𝑉) → {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V)
10 simpr 476 . . 3 (((𝐴𝑉𝐵𝑉) ∧ 𝑓(Walks‘𝐺)𝑝) → 𝑓(Walks‘𝐺)𝑝)
11 eqeq2 2662 . . . 4 (𝑎 = 𝐴 → ((𝑝‘0) = 𝑎 ↔ (𝑝‘0) = 𝐴))
12 eqeq2 2662 . . . 4 (𝑏 = 𝐵 → ((𝑝‘(#‘𝑓)) = 𝑏 ↔ (𝑝‘(#‘𝑓)) = 𝐵))
1311, 12bi2anan9 935 . . 3 ((𝑎 = 𝐴𝑏 = 𝐵) → (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ↔ ((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)))
14 biidd 252 . . 3 (𝑔 = 𝐺 → (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ↔ ((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)))
15 df-wlkson 26552 . . . 4 WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)}))
16 eqid 2651 . . . . . 6 (Vtx‘𝑔) = (Vtx‘𝑔)
17 3anass 1059 . . . . . . . 8 ((𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ↔ (𝑓(Walks‘𝑔)𝑝 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)))
18 ancom 465 . . . . . . . 8 ((𝑓(Walks‘𝑔)𝑝 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)) ↔ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝))
1917, 18bitri 264 . . . . . . 7 ((𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ↔ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝))
2019opabbii 4750 . . . . . 6 {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)} = {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝)}
2116, 16, 20mpt2eq123i 6760 . . . . 5 (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)}) = (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝)})
2221mpteq2i 4774 . . . 4 (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)})) = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝)}))
2315, 22eqtri 2673 . . 3 WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝)}))
243, 5, 7, 9, 10, 13, 14, 23mptmpt2opabbrd 7293 . 2 ((𝐴𝑉𝐵𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝)})
25 ancom 465 . . . 4 ((((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ ((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)))
26 3anass 1059 . . . 4 ((𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ ((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)))
2725, 26bitr4i 267 . . 3 ((((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵))
2827opabbii 4750 . 2 {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)}
2924, 28syl6eq 2701 1 ((𝐴𝑉𝐵𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231   class class class wbr 4685  {copab 4745  cmpt 4762  cfv 5926  (class class class)co 6690  cmpt2 6692  0cc0 9974  #chash 13157  Vtxcvtx 25919  Walkscwlks 26548  WalksOncwlkson 26549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-wlks 26551  df-wlkson 26552
This theorem is referenced by:  iswlkon  26609  wlkonprop  26610
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