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Theorem wksfval 28897
Description: The set of walks (in an undirected graph). (Contributed by AV, 30-Dec-2020.)
Hypotheses
Ref Expression
wksfval.v 𝑉 = (Vtxβ€˜πΊ)
wksfval.i 𝐼 = (iEdgβ€˜πΊ)
Assertion
Ref Expression
wksfval (𝐺 ∈ π‘Š β†’ (Walksβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(π‘“β€˜π‘˜))))})
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 wksfval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 df-wlks 28887 . 2 Walks = (𝑔 ∈ V ↦ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓 ∈ Word dom (iEdgβ€˜π‘”) ∧ 𝑝:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜π‘”) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), ((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜))))})
2 fveq2 6892 . . . . . . . 8 (𝑔 = 𝐺 β†’ (iEdgβ€˜π‘”) = (iEdgβ€˜πΊ))
3 wksfval.i . . . . . . . 8 𝐼 = (iEdgβ€˜πΊ)
42, 3eqtr4di 2791 . . . . . . 7 (𝑔 = 𝐺 β†’ (iEdgβ€˜π‘”) = 𝐼)
54dmeqd 5906 . . . . . 6 (𝑔 = 𝐺 β†’ dom (iEdgβ€˜π‘”) = dom 𝐼)
6 wrdeq 14486 . . . . . 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 2820 . . . 4 (𝑔 = 𝐺 β†’ (𝑓 ∈ Word dom (iEdgβ€˜π‘”) ↔ 𝑓 ∈ Word dom 𝐼))
9 fveq2 6892 . . . . . 6 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ))
10 wksfval.v . . . . . 6 𝑉 = (Vtxβ€˜πΊ)
119, 10eqtr4di 2791 . . . . 5 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = 𝑉)
1211feq3d 6705 . . . 4 (𝑔 = 𝐺 β†’ (𝑝:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜π‘”) ↔ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰))
134fveq1d 6894 . . . . . . 7 (𝑔 = 𝐺 β†’ ((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)) = (πΌβ€˜(π‘“β€˜π‘˜)))
1413eqeq1d 2735 . . . . . 6 (𝑔 = 𝐺 β†’ (((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)} ↔ (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}))
1513sseq2d 4015 . . . . . 6 (𝑔 = 𝐺 β†’ ({(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)) ↔ {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(π‘“β€˜π‘˜))))
1614, 15ifpbi23d 1081 . . . . 5 (𝑔 = 𝐺 β†’ (if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), ((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜))) ↔ if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(π‘“β€˜π‘˜)))))
1716ralbidv 3178 . . . 4 (𝑔 = 𝐺 β†’ (βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), ((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜))) ↔ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(π‘“β€˜π‘˜)))))
188, 12, 173anbi123d 1437 . . 3 (𝑔 = 𝐺 β†’ ((𝑓 ∈ Word dom (iEdgβ€˜π‘”) ∧ 𝑝:(0...(β™―β€˜π‘“))⟢(Vtxβ€˜π‘”) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), ((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)))) ↔ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(π‘“β€˜π‘˜))))))
1918opabbidv 5215 . 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‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), ((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜π‘”)β€˜(π‘“β€˜π‘˜))))} = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(π‘“β€˜π‘˜))))})
20 elex 3493 . 2 (𝐺 ∈ π‘Š β†’ 𝐺 ∈ V)
21 3anass 1096 . . . 4 ((𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(π‘“β€˜π‘˜)))) ↔ (𝑓 ∈ Word dom 𝐼 ∧ (𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(π‘“β€˜π‘˜))))))
2221opabbii 5216 . . 3 {βŸ¨π‘“, π‘βŸ© ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(π‘“β€˜π‘˜))))} = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓 ∈ Word dom 𝐼 ∧ (𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(π‘“β€˜π‘˜)))))}
233fvexi 6906 . . . . . 6 𝐼 ∈ V
2423dmex 7902 . . . . 5 dom 𝐼 ∈ V
25 wrdexg 14474 . . . . 5 (dom 𝐼 ∈ V β†’ Word dom 𝐼 ∈ V)
2624, 25mp1i 13 . . . 4 (𝐺 ∈ π‘Š β†’ Word dom 𝐼 ∈ V)
27 ovex 7442 . . . . . 6 (0...(β™―β€˜π‘“)) ∈ V
2810fvexi 6906 . . . . . . 7 𝑉 ∈ V
2928a1i 11 . . . . . 6 ((𝐺 ∈ π‘Š ∧ 𝑓 ∈ Word dom 𝐼) β†’ 𝑉 ∈ V)
30 mapex 8826 . . . . . 6 (((0...(β™―β€˜π‘“)) ∈ V ∧ 𝑉 ∈ V) β†’ {𝑝 ∣ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰} ∈ V)
3127, 29, 30sylancr 588 . . . . 5 ((𝐺 ∈ π‘Š ∧ 𝑓 ∈ Word dom 𝐼) β†’ {𝑝 ∣ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰} ∈ V)
32 simpl 484 . . . . . . 7 ((𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(π‘“β€˜π‘˜)))) β†’ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰)
3332ss2abi 4064 . . . . . 6 {𝑝 ∣ (𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(π‘“β€˜π‘˜))))} βŠ† {𝑝 ∣ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰}
3433a1i 11 . . . . 5 ((𝐺 ∈ π‘Š ∧ 𝑓 ∈ Word dom 𝐼) β†’ {𝑝 ∣ (𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(π‘“β€˜π‘˜))))} βŠ† {𝑝 ∣ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰})
3531, 34ssexd 5325 . . . 4 ((𝐺 ∈ π‘Š ∧ 𝑓 ∈ Word dom 𝐼) β†’ {𝑝 ∣ (𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(π‘“β€˜π‘˜))))} ∈ V)
3626, 35opabex3d 7952 . . 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‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(π‘“β€˜π‘˜)))))} ∈ V)
3722, 36eqeltrid 2838 . 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‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(π‘“β€˜π‘˜))))} ∈ V)
381, 19, 20, 37fvmptd3 7022 1 (𝐺 ∈ π‘Š β†’ (Walksβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(β™―β€˜π‘“))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π‘“))if-((π‘β€˜π‘˜) = (π‘β€˜(π‘˜ + 1)), (πΌβ€˜(π‘“β€˜π‘˜)) = {(π‘β€˜π‘˜)}, {(π‘β€˜π‘˜), (π‘β€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(π‘“β€˜π‘˜))))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397  if-wif 1062   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3062  Vcvv 3475   βŠ† wss 3949  {csn 4629  {cpr 4631  {copab 5211  dom cdm 5677  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111   + caddc 11113  ...cfz 13484  ..^cfzo 13627  β™―chash 14290  Word cword 14464  Vtxcvtx 28287  iEdgciedg 28288  Walkscwlks 28884
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-wlks 28887
This theorem is referenced by:  iswlk  28898  wlkprop  28899  wlkv  28900
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