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Theorem wlkonprop 28912
Description: Properties of a walk between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 31-Dec-2020.) (Proof shortened by AV, 16-Jan-2021.)
Hypothesis
Ref Expression
wlkson.v 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
wlkonprop (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)))
Proof of Theorem wlkonprop
Dummy variables π‘Ž 𝑏 𝑓 𝑔 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wlkson.v . . . . . 6 𝑉 = (Vtxβ€˜πΊ)
21fvexi 6905 . . . . 5 𝑉 ∈ V
3 df-wlkson 28854 . . . . . 6 WalksOn = (𝑔 ∈ V ↦ (π‘Ž ∈ (Vtxβ€˜π‘”), 𝑏 ∈ (Vtxβ€˜π‘”) ↦ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Walksβ€˜π‘”)𝑝 ∧ (π‘β€˜0) = π‘Ž ∧ (π‘β€˜(β™―β€˜π‘“)) = 𝑏)}))
41wlkson 28910 . . . . . . 7 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(WalksOnβ€˜πΊ)𝐡) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = 𝐴 ∧ (π‘β€˜(β™―β€˜π‘“)) = 𝐡)})
543adant1 1130 . . . . . 6 ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(WalksOnβ€˜πΊ)𝐡) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = 𝐴 ∧ (π‘β€˜(β™―β€˜π‘“)) = 𝐡)})
6 fveq2 6891 . . . . . . 7 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ))
76, 1eqtr4di 2790 . . . . . 6 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = 𝑉)
8 fveq2 6891 . . . . . . . 8 (𝑔 = 𝐺 β†’ (Walksβ€˜π‘”) = (Walksβ€˜πΊ))
98breqd 5159 . . . . . . 7 (𝑔 = 𝐺 β†’ (𝑓(Walksβ€˜π‘”)𝑝 ↔ 𝑓(Walksβ€˜πΊ)𝑝))
1093anbi1d 1440 . . . . . 6 (𝑔 = 𝐺 β†’ ((𝑓(Walksβ€˜π‘”)𝑝 ∧ (π‘β€˜0) = π‘Ž ∧ (π‘β€˜(β™―β€˜π‘“)) = 𝑏) ↔ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = π‘Ž ∧ (π‘β€˜(β™―β€˜π‘“)) = 𝑏)))
113, 5, 7, 7, 10bropfvvvv 8077 . . . . 5 ((𝑉 ∈ V ∧ 𝑉 ∈ V) β†’ (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ (𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))))
122, 2, 11mp2an 690 . . . 4 (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ (𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
13 3anass 1095 . . . . . 6 ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ↔ (𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)))
1413anbi1i 624 . . . . 5 (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ↔ ((𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
15 df-3an 1089 . . . . 5 ((𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ↔ ((𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
1614, 15bitr4i 277 . . . 4 (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ↔ (𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
1712, 16sylibr 233 . . 3 (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
181iswlkon 28911 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) β†’ (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 ↔ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)))
19183adantl1 1166 . . . . 5 (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) β†’ (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 ↔ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)))
2019biimpd 228 . . . 4 (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) β†’ (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)))
2120imdistani 569 . . 3 ((((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ 𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃) β†’ (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)))
2217, 21mpancom 686 . 2 (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)))
23 df-3an 1089 . 2 (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)) ↔ (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)))
2422, 23sylibr 233 1 (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474   class class class wbr 5148  {copab 5210  β€˜cfv 6543  (class class class)co 7408  0cc0 11109  β™―chash 14289  Vtxcvtx 28253  Walkscwlks 28850  WalksOncwlkson 28851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-wlkson 28854
This theorem is referenced by:  wlkoniswlk  28915  wlksoneq1eq2  28918  wlkonl1iedg  28919  wlkon2n0  28920  spthonepeq  29006  uhgrwkspth  29009  usgr2wlkspth  29013
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