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Theorem isupwlk 47276
Description: Properties of a pair of functions to be a simple walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.)
Hypotheses
Ref Expression
upwlksfval.v 𝑉 = (Vtxβ€˜πΊ)
upwlksfval.i 𝐼 = (iEdgβ€˜πΊ)
Assertion
Ref Expression
isupwlk ((𝐺 ∈ π‘Š ∧ 𝐹 ∈ π‘ˆ ∧ 𝑃 ∈ 𝑍) β†’ (𝐹(UPWalksβ€˜πΊ)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))})))
Distinct variable groups:   π‘˜,𝐺   π‘˜,𝐹   𝑃,π‘˜
Allowed substitution hints:   π‘ˆ(π‘˜)   𝐼(π‘˜)   𝑉(π‘˜)   π‘Š(π‘˜)   𝑍(π‘˜)

Proof of Theorem isupwlk
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 5153 . . 3 (𝐹(UPWalksβ€˜πΊ)𝑃 ↔ ⟨𝐹, π‘ƒβŸ© ∈ (UPWalksβ€˜πΊ))
2 upwlksfval.v . . . . . 6 𝑉 = (Vtxβ€˜πΊ)
3 upwlksfval.i . . . . . 6 𝐼 = (iEdgβ€˜πΊ)
42, 3upwlksfval 47275 . . . . 5 (𝐺 ∈ π‘Š β†’ (UPWalksβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))(πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))})})
543ad2ant1 1130 . . . 4 ((𝐺 ∈ π‘Š ∧ 𝐹 ∈ π‘ˆ ∧ 𝑃 ∈ 𝑍) β†’ (UPWalksβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))(πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))})})
65eleq2d 2815 . . 3 ((𝐺 ∈ π‘Š ∧ 𝐹 ∈ π‘ˆ ∧ 𝑃 ∈ 𝑍) β†’ (⟨𝐹, π‘ƒβŸ© ∈ (UPWalksβ€˜πΊ) ↔ ⟨𝐹, π‘ƒβŸ© ∈ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))(πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))})}))
71, 6bitrid 282 . 2 ((𝐺 ∈ π‘Š ∧ 𝐹 ∈ π‘ˆ ∧ 𝑃 ∈ 𝑍) β†’ (𝐹(UPWalksβ€˜πΊ)𝑃 ↔ ⟨𝐹, π‘ƒβŸ© ∈ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))(πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))})}))
8 eleq1 2817 . . . . . 6 (𝑓 = 𝐹 β†’ (𝑓 ∈ Word dom 𝐼 ↔ 𝐹 ∈ Word dom 𝐼))
98adantr 479 . . . . 5 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (𝑓 ∈ Word dom 𝐼 ↔ 𝐹 ∈ Word dom 𝐼))
10 simpr 483 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ 𝑝 = 𝑃)
11 fveq2 6902 . . . . . . . 8 (𝑓 = 𝐹 β†’ (β™―β€˜π‘“) = (β™―β€˜πΉ))
1211oveq2d 7442 . . . . . . 7 (𝑓 = 𝐹 β†’ (0...(β™―β€˜π‘“)) = (0...(β™―β€˜πΉ)))
1312adantr 479 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (0...(β™―β€˜π‘“)) = (0...(β™―β€˜πΉ)))
1410, 13feq12d 6715 . . . . 5 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ↔ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰))
1511oveq2d 7442 . . . . . . 7 (𝑓 = 𝐹 β†’ (0..^(β™―β€˜π‘“)) = (0..^(β™―β€˜πΉ)))
1615adantr 479 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (0..^(β™―β€˜π‘“)) = (0..^(β™―β€˜πΉ)))
17 fveq1 6901 . . . . . . . 8 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘˜) = (πΉβ€˜π‘˜))
1817fveq2d 6906 . . . . . . 7 (𝑓 = 𝐹 β†’ (πΌβ€˜(π‘“β€˜π‘˜)) = (πΌβ€˜(πΉβ€˜π‘˜)))
19 fveq1 6901 . . . . . . . 8 (𝑝 = 𝑃 β†’ (π‘β€˜π‘˜) = (π‘ƒβ€˜π‘˜))
20 fveq1 6901 . . . . . . . 8 (𝑝 = 𝑃 β†’ (π‘β€˜(π‘˜ + 1)) = (π‘ƒβ€˜(π‘˜ + 1)))
2119, 20preq12d 4750 . . . . . . 7 (𝑝 = 𝑃 β†’ {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))})
2218, 21eqeqan12d 2742 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ ((πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} ↔ (πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}))
2316, 22raleqbidv 3340 . . . . 5 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))(πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} ↔ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}))
249, 14, 233anbi123d 1432 . . . 4 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ ((𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))(πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))}) ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))})))
2524opelopabga 5539 . . 3 ((𝐹 ∈ π‘ˆ ∧ 𝑃 ∈ 𝑍) β†’ (⟨𝐹, π‘ƒβŸ© ∈ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))(πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))})} ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))})))
26253adant1 1127 . 2 ((𝐺 ∈ π‘Š ∧ 𝐹 ∈ π‘ˆ ∧ 𝑃 ∈ 𝑍) β†’ (⟨𝐹, π‘ƒβŸ© ∈ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))(πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))})} ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))})))
277, 26bitrd 278 1 ((𝐺 ∈ π‘Š ∧ 𝐹 ∈ π‘ˆ ∧ 𝑃 ∈ 𝑍) β†’ (𝐹(UPWalksβ€˜πΊ)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))})))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  {cpr 4634  βŸ¨cop 4638   class class class wbr 5152  {copab 5214  dom cdm 5682  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426  0cc0 11146  1c1 11147   + caddc 11149  ...cfz 13524  ..^cfzo 13667  β™―chash 14329  Word cword 14504  Vtxcvtx 28829  iEdgciedg 28830  UPWalkscupwlks 47273
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-fzo 13668  df-hash 14330  df-word 14505  df-upwlks 47274
This theorem is referenced by:  isupwlkg  47277  upwlkwlk  47279  upgrwlkupwlk  47280  upgrisupwlkALT  47282
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