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Theorem wlkson 27003
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 26350 . . . 4 (𝐴𝑉𝐺 ∈ V)
32adantr 474 . . 3 ((𝐴𝑉𝐵𝑉) → 𝐺 ∈ V)
4 simpl 476 . . . 4 ((𝐴𝑉𝐵𝑉) → 𝐴𝑉)
54, 1syl6eleq 2868 . . 3 ((𝐴𝑉𝐵𝑉) → 𝐴 ∈ (Vtx‘𝐺))
6 simpr 479 . . . 4 ((𝐴𝑉𝐵𝑉) → 𝐵𝑉)
76, 1syl6eleq 2868 . . 3 ((𝐴𝑉𝐵𝑉) → 𝐵 ∈ (Vtx‘𝐺))
8 wksv 26967 . . . 4 {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V
98a1i 11 . . 3 ((𝐴𝑉𝐵𝑉) → {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V)
10 simpr 479 . . 3 (((𝐴𝑉𝐵𝑉) ∧ 𝑓(Walks‘𝐺)𝑝) → 𝑓(Walks‘𝐺)𝑝)
11 eqeq2 2788 . . . 4 (𝑎 = 𝐴 → ((𝑝‘0) = 𝑎 ↔ (𝑝‘0) = 𝐴))
12 eqeq2 2788 . . . 4 (𝑏 = 𝐵 → ((𝑝‘(♯‘𝑓)) = 𝑏 ↔ (𝑝‘(♯‘𝑓)) = 𝐵))
1311, 12bi2anan9 629 . . 3 ((𝑎 = 𝐴𝑏 = 𝐵) → (((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ↔ ((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵)))
14 biidd 254 . . 3 (𝑔 = 𝐺 → (((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ↔ ((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)))
15 df-wlkson 26948 . . . 4 WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)}))
16 eqid 2777 . . . . . 6 (Vtx‘𝑔) = (Vtx‘𝑔)
17 3anass 1079 . . . . . . . 8 ((𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ↔ (𝑓(Walks‘𝑔)𝑝 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)))
18 ancom 454 . . . . . . . 8 ((𝑓(Walks‘𝑔)𝑝 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)) ↔ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝))
1917, 18bitri 267 . . . . . . 7 ((𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ↔ (((𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏) ∧ 𝑓(Walks‘𝑔)𝑝))
2019opabbii 4953 . . . . . 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 6995 . . . . 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 4976 . . . 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 2801 . . 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 7528 . 2 ((𝐴𝑉𝐵𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝)})
25 ancom 454 . . . 4 ((((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ ((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵)))
26 3anass 1079 . . . 4 ((𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ ((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵)))
2725, 26bitr4i 270 . . 3 ((((𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵) ∧ 𝑓(Walks‘𝐺)𝑝) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵))
2827opabbii 4953 . 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 2829 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 386  w3a 1071   = wceq 1601  wcel 2106  Vcvv 3397   class class class wbr 4886  {copab 4948  cmpt 4965  cfv 6135  (class class class)co 6922  cmpt2 6924  0cc0 10272  chash 13435  Vtxcvtx 26344  Walkscwlks 26944  WalksOncwlkson 26945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ifp 1047  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-hash 13436  df-word 13600  df-wlks 26947  df-wlkson 26948
This theorem is referenced by:  iswlkon  27004  wlkonprop  27005
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