MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wlkonprop Structured version   Visualization version   GIF version

Theorem wlkonprop 28655
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 6860 . . . . 5 𝑉 ∈ V
3 df-wlkson 28597 . . . . . 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 28653 . . . . . . 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 1131 . . . . . 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 6846 . . . . . . 7 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ))
76, 1eqtr4di 2791 . . . . . 6 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = 𝑉)
8 fveq2 6846 . . . . . . . 8 (𝑔 = 𝐺 β†’ (Walksβ€˜π‘”) = (Walksβ€˜πΊ))
98breqd 5120 . . . . . . 7 (𝑔 = 𝐺 β†’ (𝑓(Walksβ€˜π‘”)𝑝 ↔ 𝑓(Walksβ€˜πΊ)𝑝))
1093anbi1d 1441 . . . . . 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 8028 . . . . 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 691 . . . 4 (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ (𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
13 3anass 1096 . . . . . 6 ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ↔ (𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)))
1413anbi1i 625 . . . . 5 (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ↔ ((𝐺 ∈ V ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
15 df-3an 1090 . . . . 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 28654 . . . . . 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 1167 . . . . 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 570 . . 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 687 . 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 1090 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3447   class class class wbr 5109  {copab 5171  β€˜cfv 6500  (class class class)co 7361  0cc0 11059  β™―chash 14239  Vtxcvtx 27996  Walkscwlks 28593  WalksOncwlkson 28594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-wlkson 28597
This theorem is referenced by:  wlkoniswlk  28658  wlksoneq1eq2  28661  wlkonl1iedg  28662  wlkon2n0  28663  spthonepeq  28749  uhgrwkspth  28752  usgr2wlkspth  28756
  Copyright terms: Public domain W3C validator