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Theorem upgr2wlk 28963
Description: Properties of a pair of functions to be a walk of length 2 in a pseudograph. Note that the vertices need not to be distinct and the edges can be loops or multiedges. (Contributed by Alexander van der Vekens, 16-Feb-2018.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 28-Oct-2021.)
Hypotheses
Ref Expression
upgr2wlk.v 𝑉 = (Vtxβ€˜πΊ)
upgr2wlk.i 𝐼 = (iEdgβ€˜πΊ)
Assertion
Ref Expression
upgr2wlk (𝐺 ∈ UPGraph β†’ ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 2) ↔ (𝐹:(0..^2)⟢dom 𝐼 ∧ 𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))))
Proof of Theorem upgr2wlk
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 upgr2wlk.v . . . 4 𝑉 = (Vtxβ€˜πΊ)
2 upgr2wlk.i . . . 4 𝐼 = (iEdgβ€˜πΊ)
31, 2upgriswlk 28936 . . 3 (𝐺 ∈ UPGraph β†’ (𝐹(Walksβ€˜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))})))
43anbi1d 630 . 2 (𝐺 ∈ UPGraph β†’ ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 2) ↔ ((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))}) ∧ (β™―β€˜πΉ) = 2)))
5 iswrdb 14472 . . . . . . . . 9 (𝐹 ∈ Word dom 𝐼 ↔ 𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐼)
6 oveq2 7419 . . . . . . . . . 10 ((β™―β€˜πΉ) = 2 β†’ (0..^(β™―β€˜πΉ)) = (0..^2))
76feq2d 6703 . . . . . . . . 9 ((β™―β€˜πΉ) = 2 β†’ (𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐼 ↔ 𝐹:(0..^2)⟢dom 𝐼))
85, 7bitrid 282 . . . . . . . 8 ((β™―β€˜πΉ) = 2 β†’ (𝐹 ∈ Word dom 𝐼 ↔ 𝐹:(0..^2)⟢dom 𝐼))
9 oveq2 7419 . . . . . . . . 9 ((β™―β€˜πΉ) = 2 β†’ (0...(β™―β€˜πΉ)) = (0...2))
109feq2d 6703 . . . . . . . 8 ((β™―β€˜πΉ) = 2 β†’ (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ↔ 𝑃:(0...2)βŸΆπ‘‰))
11 fzo0to2pr 13719 . . . . . . . . . . 11 (0..^2) = {0, 1}
126, 11eqtrdi 2788 . . . . . . . . . 10 ((β™―β€˜πΉ) = 2 β†’ (0..^(β™―β€˜πΉ)) = {0, 1})
1312raleqdv 3325 . . . . . . . . 9 ((β™―β€˜πΉ) = 2 β†’ (βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} ↔ βˆ€π‘˜ ∈ {0, 1} (πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}))
14 2wlklem 28962 . . . . . . . . 9 (βˆ€π‘˜ ∈ {0, 1} (πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} ↔ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
1513, 14bitrdi 286 . . . . . . . 8 ((β™―β€˜πΉ) = 2 β†’ (βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} ↔ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})))
168, 10, 153anbi123d 1436 . . . . . . 7 ((β™―β€˜πΉ) = 2 β†’ ((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))}) ↔ (𝐹:(0..^2)⟢dom 𝐼 ∧ 𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))))
1716adantl 482 . . . . . 6 ((𝐺 ∈ UPGraph ∧ (β™―β€˜πΉ) = 2) β†’ ((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))}) ↔ (𝐹:(0..^2)⟢dom 𝐼 ∧ 𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))))
18 3anass 1095 . . . . . 6 ((𝐹:(0..^2)⟢dom 𝐼 ∧ 𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ↔ (𝐹:(0..^2)⟢dom 𝐼 ∧ (𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))))
1917, 18bitrdi 286 . . . . 5 ((𝐺 ∈ UPGraph ∧ (β™―β€˜πΉ) = 2) β†’ ((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))}) ↔ (𝐹:(0..^2)⟢dom 𝐼 ∧ (𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})))))
2019ex 413 . . . 4 (𝐺 ∈ UPGraph β†’ ((β™―β€˜πΉ) = 2 β†’ ((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))}) ↔ (𝐹:(0..^2)⟢dom 𝐼 ∧ (𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))))))
2120pm5.32rd 578 . . 3 (𝐺 ∈ UPGraph β†’ (((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))}) ∧ (β™―β€˜πΉ) = 2) ↔ ((𝐹:(0..^2)⟢dom 𝐼 ∧ (𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))) ∧ (β™―β€˜πΉ) = 2)))
22 3anass 1095 . . . 4 (((𝐹:(0..^2)⟢dom 𝐼 ∧ (β™―β€˜πΉ) = 2) ∧ 𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ↔ ((𝐹:(0..^2)⟢dom 𝐼 ∧ (β™―β€˜πΉ) = 2) ∧ (𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))))
23 an32 644 . . . 4 (((𝐹:(0..^2)⟢dom 𝐼 ∧ (β™―β€˜πΉ) = 2) ∧ (𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))) ↔ ((𝐹:(0..^2)⟢dom 𝐼 ∧ (𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))) ∧ (β™―β€˜πΉ) = 2))
2422, 23bitri 274 . . 3 (((𝐹:(0..^2)⟢dom 𝐼 ∧ (β™―β€˜πΉ) = 2) ∧ 𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ↔ ((𝐹:(0..^2)⟢dom 𝐼 ∧ (𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))) ∧ (β™―β€˜πΉ) = 2))
2521, 24bitr4di 288 . 2 (𝐺 ∈ UPGraph β†’ (((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))}) ∧ (β™―β€˜πΉ) = 2) ↔ ((𝐹:(0..^2)⟢dom 𝐼 ∧ (β™―β€˜πΉ) = 2) ∧ 𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))))
26 2nn0 12491 . . . . . . 7 2 ∈ β„•0
27 fnfzo0hash 14411 . . . . . . 7 ((2 ∈ β„•0 ∧ 𝐹:(0..^2)⟢dom 𝐼) β†’ (β™―β€˜πΉ) = 2)
2826, 27mpan 688 . . . . . 6 (𝐹:(0..^2)⟢dom 𝐼 β†’ (β™―β€˜πΉ) = 2)
2928pm4.71i 560 . . . . 5 (𝐹:(0..^2)⟢dom 𝐼 ↔ (𝐹:(0..^2)⟢dom 𝐼 ∧ (β™―β€˜πΉ) = 2))
3029bicomi 223 . . . 4 ((𝐹:(0..^2)⟢dom 𝐼 ∧ (β™―β€˜πΉ) = 2) ↔ 𝐹:(0..^2)⟢dom 𝐼)
3130a1i 11 . . 3 (𝐺 ∈ UPGraph β†’ ((𝐹:(0..^2)⟢dom 𝐼 ∧ (β™―β€˜πΉ) = 2) ↔ 𝐹:(0..^2)⟢dom 𝐼))
32313anbi1d 1440 . 2 (𝐺 ∈ UPGraph β†’ (((𝐹:(0..^2)⟢dom 𝐼 ∧ (β™―β€˜πΉ) = 2) ∧ 𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ↔ (𝐹:(0..^2)⟢dom 𝐼 ∧ 𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))))
334, 25, 323bitrd 304 1 (𝐺 ∈ UPGraph β†’ ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 2) ↔ (𝐹:(0..^2)⟢dom 𝐼 ∧ 𝑃:(0...2)βŸΆπ‘‰ ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {cpr 4630   class class class wbr 5148  dom cdm 5676  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  0cc0 11112  1c1 11113   + caddc 11115  2c2 12269  β„•0cn0 12474  ...cfz 13486  ..^cfzo 13629  β™―chash 14292  Word cword 14466  Vtxcvtx 28294  iEdgciedg 28295  UPGraphcupgr 28378  Walkscwlks 28891
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 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  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-nel 3047  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-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  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-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  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-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-n0 12475  df-xnn0 12547  df-z 12561  df-uz 12825  df-fz 13487  df-fzo 13630  df-hash 14293  df-word 14467  df-edg 28346  df-uhgr 28356  df-upgr 28380  df-wlks 28894
This theorem is referenced by:  umgrwwlks2on  29249
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