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Theorem wwlksnextprop 28857
Description: Adding additional properties to the set of walks (as words) of a fixed length starting at a fixed vertex. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 29-Oct-2022.)
Hypotheses
Ref Expression
wwlksnextprop.x 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
wwlksnextprop.e 𝐸 = (Edgβ€˜πΊ)
wwlksnextprop.y π‘Œ = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}
Assertion
Ref Expression
wwlksnextprop (𝑁 ∈ β„•0 β†’ {π‘₯ ∈ 𝑋 ∣ (π‘₯β€˜0) = 𝑃} = {π‘₯ ∈ 𝑋 ∣ βˆƒπ‘¦ ∈ π‘Œ ((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸)})
Distinct variable groups:   𝑀,𝐺   𝑀,𝑁   𝑀,𝑃   𝑦,𝐸   π‘₯,𝑁,𝑦   𝑦,𝑃   𝑦,𝑋   𝑦,π‘Œ   π‘₯,𝑀
Allowed substitution hints:   𝑃(π‘₯)   𝐸(π‘₯,𝑀)   𝐺(π‘₯,𝑦)   𝑋(π‘₯,𝑀)   π‘Œ(π‘₯,𝑀)

Proof of Theorem wwlksnextprop
StepHypRef Expression
1 eqidd 2737 . . . . 5 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ (π‘₯ prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)))
2 wwlksnextprop.x . . . . . . . . 9 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
32wwlksnextproplem1 28854 . . . . . . . 8 ((π‘₯ ∈ 𝑋 ∧ 𝑁 ∈ β„•0) β†’ ((π‘₯ prefix (𝑁 + 1))β€˜0) = (π‘₯β€˜0))
43ancoms 459 . . . . . . 7 ((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯ prefix (𝑁 + 1))β€˜0) = (π‘₯β€˜0))
54adantr 481 . . . . . 6 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ ((π‘₯ prefix (𝑁 + 1))β€˜0) = (π‘₯β€˜0))
6 eqeq2 2748 . . . . . . 7 ((π‘₯β€˜0) = 𝑃 β†’ (((π‘₯ prefix (𝑁 + 1))β€˜0) = (π‘₯β€˜0) ↔ ((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃))
76adantl 482 . . . . . 6 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ (((π‘₯ prefix (𝑁 + 1))β€˜0) = (π‘₯β€˜0) ↔ ((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃))
85, 7mpbid 231 . . . . 5 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ ((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃)
9 wwlksnextprop.e . . . . . . . 8 𝐸 = (Edgβ€˜πΊ)
102, 9wwlksnextproplem2 28855 . . . . . . 7 ((π‘₯ ∈ 𝑋 ∧ 𝑁 ∈ β„•0) β†’ {(lastSβ€˜(π‘₯ prefix (𝑁 + 1))), (lastSβ€˜π‘₯)} ∈ 𝐸)
1110ancoms 459 . . . . . 6 ((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) β†’ {(lastSβ€˜(π‘₯ prefix (𝑁 + 1))), (lastSβ€˜π‘₯)} ∈ 𝐸)
1211adantr 481 . . . . 5 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ {(lastSβ€˜(π‘₯ prefix (𝑁 + 1))), (lastSβ€˜π‘₯)} ∈ 𝐸)
13 simpr 485 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
1413adantr 481 . . . . . . 7 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ π‘₯ ∈ 𝑋)
15 simpr 485 . . . . . . 7 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ (π‘₯β€˜0) = 𝑃)
16 simpll 765 . . . . . . 7 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ 𝑁 ∈ β„•0)
17 wwlksnextprop.y . . . . . . . 8 π‘Œ = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}
182, 9, 17wwlksnextproplem3 28856 . . . . . . 7 ((π‘₯ ∈ 𝑋 ∧ (π‘₯β€˜0) = 𝑃 ∧ 𝑁 ∈ β„•0) β†’ (π‘₯ prefix (𝑁 + 1)) ∈ π‘Œ)
1914, 15, 16, 18syl3anc 1371 . . . . . 6 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ (π‘₯ prefix (𝑁 + 1)) ∈ π‘Œ)
20 eqeq2 2748 . . . . . . . 8 (𝑦 = (π‘₯ prefix (𝑁 + 1)) β†’ ((π‘₯ prefix (𝑁 + 1)) = 𝑦 ↔ (π‘₯ prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1))))
21 fveq1 6841 . . . . . . . . 9 (𝑦 = (π‘₯ prefix (𝑁 + 1)) β†’ (π‘¦β€˜0) = ((π‘₯ prefix (𝑁 + 1))β€˜0))
2221eqeq1d 2738 . . . . . . . 8 (𝑦 = (π‘₯ prefix (𝑁 + 1)) β†’ ((π‘¦β€˜0) = 𝑃 ↔ ((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃))
23 fveq2 6842 . . . . . . . . . 10 (𝑦 = (π‘₯ prefix (𝑁 + 1)) β†’ (lastSβ€˜π‘¦) = (lastSβ€˜(π‘₯ prefix (𝑁 + 1))))
2423preq1d 4700 . . . . . . . . 9 (𝑦 = (π‘₯ prefix (𝑁 + 1)) β†’ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} = {(lastSβ€˜(π‘₯ prefix (𝑁 + 1))), (lastSβ€˜π‘₯)})
2524eleq1d 2822 . . . . . . . 8 (𝑦 = (π‘₯ prefix (𝑁 + 1)) β†’ ({(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸 ↔ {(lastSβ€˜(π‘₯ prefix (𝑁 + 1))), (lastSβ€˜π‘₯)} ∈ 𝐸))
2620, 22, 253anbi123d 1436 . . . . . . 7 (𝑦 = (π‘₯ prefix (𝑁 + 1)) β†’ (((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸) ↔ ((π‘₯ prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)) ∧ ((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃 ∧ {(lastSβ€˜(π‘₯ prefix (𝑁 + 1))), (lastSβ€˜π‘₯)} ∈ 𝐸)))
2726adantl 482 . . . . . 6 ((((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) ∧ 𝑦 = (π‘₯ prefix (𝑁 + 1))) β†’ (((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸) ↔ ((π‘₯ prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)) ∧ ((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃 ∧ {(lastSβ€˜(π‘₯ prefix (𝑁 + 1))), (lastSβ€˜π‘₯)} ∈ 𝐸)))
2819, 27rspcedv 3574 . . . . 5 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ (((π‘₯ prefix (𝑁 + 1)) = (π‘₯ prefix (𝑁 + 1)) ∧ ((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃 ∧ {(lastSβ€˜(π‘₯ prefix (𝑁 + 1))), (lastSβ€˜π‘₯)} ∈ 𝐸) β†’ βˆƒπ‘¦ ∈ π‘Œ ((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸)))
291, 8, 12, 28mp3and 1464 . . . 4 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ (π‘₯β€˜0) = 𝑃) β†’ βˆƒπ‘¦ ∈ π‘Œ ((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸))
3029ex 413 . . 3 ((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯β€˜0) = 𝑃 β†’ βˆƒπ‘¦ ∈ π‘Œ ((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸)))
3121eqcoms 2744 . . . . . . . . 9 ((π‘₯ prefix (𝑁 + 1)) = 𝑦 β†’ (π‘¦β€˜0) = ((π‘₯ prefix (𝑁 + 1))β€˜0))
3231eqeq1d 2738 . . . . . . . 8 ((π‘₯ prefix (𝑁 + 1)) = 𝑦 β†’ ((π‘¦β€˜0) = 𝑃 ↔ ((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃))
333eqcomd 2742 . . . . . . . . . . 11 ((π‘₯ ∈ 𝑋 ∧ 𝑁 ∈ β„•0) β†’ (π‘₯β€˜0) = ((π‘₯ prefix (𝑁 + 1))β€˜0))
3433ancoms 459 . . . . . . . . . 10 ((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯β€˜0) = ((π‘₯ prefix (𝑁 + 1))β€˜0))
3534adantr 481 . . . . . . . . 9 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (π‘₯β€˜0) = ((π‘₯ prefix (𝑁 + 1))β€˜0))
36 eqeq2 2748 . . . . . . . . . 10 (𝑃 = ((π‘₯ prefix (𝑁 + 1))β€˜0) β†’ ((π‘₯β€˜0) = 𝑃 ↔ (π‘₯β€˜0) = ((π‘₯ prefix (𝑁 + 1))β€˜0)))
3736eqcoms 2744 . . . . . . . . 9 (((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃 β†’ ((π‘₯β€˜0) = 𝑃 ↔ (π‘₯β€˜0) = ((π‘₯ prefix (𝑁 + 1))β€˜0)))
3835, 37syl5ibr 245 . . . . . . . 8 (((π‘₯ prefix (𝑁 + 1))β€˜0) = 𝑃 β†’ (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (π‘₯β€˜0) = 𝑃))
3932, 38syl6bi 252 . . . . . . 7 ((π‘₯ prefix (𝑁 + 1)) = 𝑦 β†’ ((π‘¦β€˜0) = 𝑃 β†’ (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (π‘₯β€˜0) = 𝑃)))
4039imp 407 . . . . . 6 (((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃) β†’ (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (π‘₯β€˜0) = 𝑃))
41403adant3 1132 . . . . 5 (((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸) β†’ (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (π‘₯β€˜0) = 𝑃))
4241com12 32 . . . 4 (((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸) β†’ (π‘₯β€˜0) = 𝑃))
4342rexlimdva 3152 . . 3 ((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) β†’ (βˆƒπ‘¦ ∈ π‘Œ ((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸) β†’ (π‘₯β€˜0) = 𝑃))
4430, 43impbid 211 . 2 ((𝑁 ∈ β„•0 ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯β€˜0) = 𝑃 ↔ βˆƒπ‘¦ ∈ π‘Œ ((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸)))
4544rabbidva 3414 1 (𝑁 ∈ β„•0 β†’ {π‘₯ ∈ 𝑋 ∣ (π‘₯β€˜0) = 𝑃} = {π‘₯ ∈ 𝑋 ∣ βˆƒπ‘¦ ∈ π‘Œ ((π‘₯ prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘₯)} ∈ 𝐸)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3073  {crab 3407  {cpr 4588  β€˜cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052   + caddc 11054  β„•0cn0 12413  lastSclsw 14450   prefix cpfx 14558  Edgcedg 27998   WWalksN cwwlksn 28771
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-lsw 14451  df-substr 14529  df-pfx 14559  df-wwlks 28775  df-wwlksn 28776
This theorem is referenced by: (None)
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