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Theorem upwlksfval 47308
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 47307 . 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 6891 . . . . . . . 8 (𝑔 = 𝐺 β†’ (iEdgβ€˜π‘”) = (iEdgβ€˜πΊ))
3 upwlksfval.i . . . . . . . 8 𝐼 = (iEdgβ€˜πΊ)
42, 3eqtr4di 2783 . . . . . . 7 (𝑔 = 𝐺 β†’ (iEdgβ€˜π‘”) = 𝐼)
54dmeqd 5902 . . . . . 6 (𝑔 = 𝐺 β†’ dom (iEdgβ€˜π‘”) = dom 𝐼)
6 wrdeq 14516 . . . . . 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 2811 . . . 4 (𝑔 = 𝐺 β†’ (𝑓 ∈ Word dom (iEdgβ€˜π‘”) ↔ 𝑓 ∈ Word dom 𝐼))
9 fveq2 6891 . . . . . 6 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ))
10 upwlksfval.v . . . . . 6 𝑉 = (Vtxβ€˜πΊ)
119, 10eqtr4di 2783 . . . . 5 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = 𝑉)
1211feq3d 6703 . . . 4 (𝑔 = 𝐺 β†’ (𝑝:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜π‘”) ↔ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰))
134fveq1d 6893 . . . . . 6 (𝑔 = 𝐺 β†’ ((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)) = (πΌβ€˜(π‘“β€˜π‘˜)))
1413eqeq1d 2727 . . . . 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 3168 . . . 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 5209 . 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 3482 . 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 5210 . . 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 6905 . . . . . 6 𝐼 ∈ V
2221dmex 7913 . . . . 5 dom 𝐼 ∈ V
23 wrdexg 14504 . . . . 5 (dom 𝐼 ∈ V β†’ Word dom 𝐼 ∈ V)
2422, 23mp1i 13 . . . 4 (𝐺 ∈ π‘Š β†’ Word dom 𝐼 ∈ V)
25 ovex 7448 . . . . . 6 (0...(β™―β€˜π‘“)) ∈ V
2610fvexi 6905 . . . . . . 7 𝑉 ∈ V
2726a1i 11 . . . . . 6 ((𝐺 ∈ π‘Š ∧ 𝑓 ∈ Word dom 𝐼) β†’ 𝑉 ∈ V)
28 mapex 8847 . . . . . 6 (((0...(β™―β€˜π‘“)) ∈ V ∧ 𝑉 ∈ V) β†’ {𝑝 ∣ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰} ∈ V)
2925, 27, 28sylancr 585 . . . . 5 ((𝐺 ∈ π‘Š ∧ 𝑓 ∈ Word dom 𝐼) β†’ {𝑝 ∣ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰} ∈ V)
30 simpl 481 . . . . . . 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 4055 . . . . . 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 5319 . . . 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 7965 . . 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 2829 . 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 7022 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2702  βˆ€wral 3051  Vcvv 3463   βŠ† wss 3940  {cpr 4626  {copab 5205  dom cdm 5672  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7415  0cc0 11136  1c1 11137   + caddc 11139  ...cfz 13514  ..^cfzo 13657  β™―chash 14319  Word cword 14494  Vtxcvtx 28851  iEdgciedg 28852  UPWalkscupwlks 47306
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 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-er 8721  df-map 8843  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-n0 12501  df-z 12587  df-uz 12851  df-fz 13515  df-fzo 13658  df-hash 14320  df-word 14495  df-upwlks 47307
This theorem is referenced by:  isupwlk  47309  isupwlkg  47310  upwlkbprop  47311
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