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Theorem upwlksfval 47090
Description: The set of simple walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-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
upwlksfval (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))})})
Distinct variable groups:   𝑓,𝐺,π‘˜,𝑝   𝑓,𝐼,𝑝   𝑉,𝑝   𝑓,π‘Š
Allowed substitution hints:   𝐼(π‘˜)   𝑉(𝑓,π‘˜)   π‘Š(π‘˜,𝑝)

Proof of Theorem upwlksfval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 df-upwlks 47089 . 2 UPWalks = (𝑔 ∈ V ↦ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓 ∈ Word dom (iEdgβ€˜π‘”) ∧ 𝑝:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜π‘”) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))})})
2 fveq2 6885 . . . . . . . 8 (𝑔 = 𝐺 β†’ (iEdgβ€˜π‘”) = (iEdgβ€˜πΊ))
3 upwlksfval.i . . . . . . . 8 𝐼 = (iEdgβ€˜πΊ)
42, 3eqtr4di 2784 . . . . . . 7 (𝑔 = 𝐺 β†’ (iEdgβ€˜π‘”) = 𝐼)
54dmeqd 5899 . . . . . 6 (𝑔 = 𝐺 β†’ dom (iEdgβ€˜π‘”) = dom 𝐼)
6 wrdeq 14492 . . . . . 6 (dom (iEdgβ€˜π‘”) = dom 𝐼 β†’ Word dom (iEdgβ€˜π‘”) = Word dom 𝐼)
75, 6syl 17 . . . . 5 (𝑔 = 𝐺 β†’ Word dom (iEdgβ€˜π‘”) = Word dom 𝐼)
87eleq2d 2813 . . . 4 (𝑔 = 𝐺 β†’ (𝑓 ∈ Word dom (iEdgβ€˜π‘”) ↔ 𝑓 ∈ Word dom 𝐼))
9 fveq2 6885 . . . . . 6 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ))
10 upwlksfval.v . . . . . 6 𝑉 = (Vtxβ€˜πΊ)
119, 10eqtr4di 2784 . . . . 5 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = 𝑉)
1211feq3d 6698 . . . 4 (𝑔 = 𝐺 β†’ (𝑝:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜π‘”) ↔ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰))
134fveq1d 6887 . . . . . 6 (𝑔 = 𝐺 β†’ ((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)) = (πΌβ€˜(π‘“β€˜π‘˜)))
1413eqeq1d 2728 . . . . 5 (𝑔 = 𝐺 β†’ (((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} ↔ (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))}))
1514ralbidv 3171 . . . 4 (𝑔 = 𝐺 β†’ (βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} ↔ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))(πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))}))
168, 12, 153anbi123d 1432 . . 3 (𝑔 = 𝐺 β†’ ((𝑓 ∈ Word dom (iEdgβ€˜π‘”) ∧ 𝑝:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜π‘”) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))}) ↔ (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))})))
1716opabbidv 5207 . 2 (𝑔 = 𝐺 β†’ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓 ∈ Word dom (iEdgβ€˜π‘”) ∧ 𝑝:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜π‘”) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))})} = {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))})})
18 elex 3487 . 2 (𝐺 ∈ π‘Š β†’ 𝐺 ∈ V)
19 3anass 1092 . . . 4 ((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‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))(πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))})))
2019opabbii 5208 . . 3 {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))})} = {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))}))}
213fvexi 6899 . . . . . 6 𝐼 ∈ V
2221dmex 7899 . . . . 5 dom 𝐼 ∈ V
23 wrdexg 14480 . . . . 5 (dom 𝐼 ∈ V β†’ Word dom 𝐼 ∈ V)
2422, 23mp1i 13 . . . 4 (𝐺 ∈ π‘Š β†’ Word dom 𝐼 ∈ V)
25 ovex 7438 . . . . . 6 (0...(β™―β€˜π‘“)) ∈ V
2610fvexi 6899 . . . . . . 7 𝑉 ∈ V
2726a1i 11 . . . . . 6 ((𝐺 ∈ π‘Š ∧ 𝑓 ∈ Word dom 𝐼) β†’ 𝑉 ∈ V)
28 mapex 8828 . . . . . 6 (((0...(β™―β€˜π‘“)) ∈ V ∧ 𝑉 ∈ V) β†’ {𝑝 ∣ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰} ∈ V)
2925, 27, 28sylancr 586 . . . . 5 ((𝐺 ∈ π‘Š ∧ 𝑓 ∈ Word dom 𝐼) β†’ {𝑝 ∣ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰} ∈ V)
30 simpl 482 . . . . . . 7 ((𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))(πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))}) β†’ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰)
3130ss2abi 4058 . . . . . 6 {𝑝 ∣ (𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))(πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))})} βŠ† {𝑝 ∣ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰}
3231a1i 11 . . . . 5 ((𝐺 ∈ π‘Š ∧ 𝑓 ∈ Word dom 𝐼) β†’ {𝑝 ∣ (𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))(πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))})} βŠ† {𝑝 ∣ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰})
3329, 32ssexd 5317 . . . 4 ((𝐺 ∈ π‘Š ∧ 𝑓 ∈ Word dom 𝐼) β†’ {𝑝 ∣ (𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))(πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))})} ∈ V)
3424, 33opabex3d 7951 . . 3 (𝐺 ∈ π‘Š β†’ {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))}))} ∈ V)
3520, 34eqeltrid 2831 . 2 (𝐺 ∈ π‘Š β†’ {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))})} ∈ V)
361, 17, 18, 35fvmptd3 7015 1 (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))})})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2703  βˆ€wral 3055  Vcvv 3468   βŠ† wss 3943  {cpr 4625  {copab 5203  dom cdm 5669  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405  0cc0 11112  1c1 11113   + caddc 11115  ...cfz 13490  ..^cfzo 13633  β™―chash 14295  Word cword 14470  Vtxcvtx 28764  iEdgciedg 28765  UPWalkscupwlks 47088
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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  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 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-fzo 13634  df-hash 14296  df-word 14471  df-upwlks 47089
This theorem is referenced by:  isupwlk  47091  isupwlkg  47092  upwlkbprop  47093
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