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Theorem iswlkon 29484
Description: Properties of a pair of functions to be a walk between two given vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 2-Nov-2017.) (Revised by AV, 31-Dec-2020.) (Revised by AV, 22-Mar-2021.)
Hypothesis
Ref Expression
wlkson.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
iswlkon (((𝐴𝑉𝐵𝑉) ∧ (𝐹𝑈𝑃𝑍)) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)))

Proof of Theorem iswlkon
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wlkson.v . . . 4 𝑉 = (Vtx‘𝐺)
21wlkson 29483 . . 3 ((𝐴𝑉𝐵𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵)})
3 fveq1 6896 . . . . 5 (𝑝 = 𝑃 → (𝑝‘0) = (𝑃‘0))
43adantl 481 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝‘0) = (𝑃‘0))
54eqeq1d 2730 . . 3 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑝‘0) = 𝐴 ↔ (𝑃‘0) = 𝐴))
6 simpr 484 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → 𝑝 = 𝑃)
7 fveq2 6897 . . . . . 6 (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
87adantr 480 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (♯‘𝑓) = (♯‘𝐹))
96, 8fveq12d 6904 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝‘(♯‘𝑓)) = (𝑃‘(♯‘𝐹)))
109eqeq1d 2730 . . 3 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑝‘(♯‘𝑓)) = 𝐵 ↔ (𝑃‘(♯‘𝐹)) = 𝐵))
112, 5, 102rbropap 5568 . 2 (((𝐴𝑉𝐵𝑉) ∧ 𝐹𝑈𝑃𝑍) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)))
12113expb 1118 1 (((𝐴𝑉𝐵𝑉) ∧ (𝐹𝑈𝑃𝑍)) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099   class class class wbr 5148  cfv 6548  (class class class)co 7420  0cc0 11139  chash 14322  Vtxcvtx 28822  Walkscwlks 29423  WalksOncwlkson 29424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-wlkson 29427
This theorem is referenced by:  wlkonprop  29485  wlkonwlk  29489  wlkonwlk1l  29490  isspthonpth  29576  2wlkond  29761  umgr2adedgwlkonALT  29771  umgr2wlkon  29774  wpthswwlks2on  29785  0wlkon  29943  1pthond  29967
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