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Theorem wlkonprop 29409
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 6896 . . . . 5 𝑉 ∈ V
3 df-wlkson 29351 . . . . . 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 29407 . . . . . . 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 1127 . . . . . 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 6882 . . . . . . 7 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ))
76, 1eqtr4di 2782 . . . . . 6 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = 𝑉)
8 fveq2 6882 . . . . . . . 8 (𝑔 = 𝐺 β†’ (Walksβ€˜π‘”) = (Walksβ€˜πΊ))
98breqd 5150 . . . . . . 7 (𝑔 = 𝐺 β†’ (𝑓(Walksβ€˜π‘”)𝑝 ↔ 𝑓(Walksβ€˜πΊ)𝑝))
1093anbi1d 1436 . . . . . 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 8073 . . . . 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 689 . . . 4 (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ (𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
13 3anass 1092 . . . . . 6 ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ↔ (𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)))
1413anbi1i 623 . . . . 5 (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ↔ ((𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
15 df-3an 1086 . . . . 5 ((𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ↔ ((𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
1614, 15bitr4i 278 . . . 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 29408 . . . . . 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 1163 . . . . 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 568 . . 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 685 . 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 1086 . 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3466   class class class wbr 5139  {copab 5201  β€˜cfv 6534  (class class class)co 7402  0cc0 11107  β™―chash 14291  Vtxcvtx 28749  Walkscwlks 29347  WalksOncwlkson 29348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-wlkson 29351
This theorem is referenced by:  wlkoniswlk  29412  wlksoneq1eq2  29415  wlkonl1iedg  29416  wlkon2n0  29417  spthonepeq  29503  uhgrwkspth  29506  usgr2wlkspth  29510
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