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Theorem iswwlksnon 29107
Description: The set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 14-Mar-2022.)
Hypothesis
Ref Expression
iswwlksnon.v 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
iswwlksnon (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)}
Distinct variable groups:   𝑀,𝐴   𝑀,𝐡   𝑀,𝐺   𝑀,𝑁   𝑀,𝑉

Proof of Theorem iswwlksnon
Dummy variables π‘Ž 𝑏 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ov 7446 . . 3 (π΄βˆ…π΅) = βˆ…
2 df-wwlksnon 29086 . . . . 5 WWalksNOn = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ (π‘Ž ∈ (Vtxβ€˜π‘”), 𝑏 ∈ (Vtxβ€˜π‘”) ↦ {𝑀 ∈ (𝑛 WWalksN 𝑔) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘›) = 𝑏)}))
32mpondm0 7647 . . . 4 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝑁 WWalksNOn 𝐺) = βˆ…)
43oveqd 7426 . . 3 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = (π΄βˆ…π΅))
5 df-wwlksn 29085 . . . . . 6 WWalksN = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ {𝑀 ∈ (WWalksβ€˜π‘”) ∣ (β™―β€˜π‘€) = (𝑛 + 1)})
65mpondm0 7647 . . . . 5 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝑁 WWalksN 𝐺) = βˆ…)
76rabeqdv 3448 . . . 4 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)} = {𝑀 ∈ βˆ… ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)})
8 rab0 4383 . . . 4 {𝑀 ∈ βˆ… ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)} = βˆ…
97, 8eqtrdi 2789 . . 3 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)} = βˆ…)
101, 4, 93eqtr4a 2799 . 2 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)})
11 iswwlksnon.v . . . . . . . . 9 𝑉 = (Vtxβ€˜πΊ)
1211wwlksnon 29105 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝑁 WWalksNOn 𝐺) = (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)}))
1312adantr 482 . . . . . . 7 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝑁 WWalksNOn 𝐺) = (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)}))
1413oveqd 7426 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = (𝐴(π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)})𝐡))
15 eqid 2733 . . . . . . . 8 (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)}) = (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)})
1615mpondm0 7647 . . . . . . 7 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)})𝐡) = βˆ…)
1716adantl 483 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝐴(π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)})𝐡) = βˆ…)
1814, 17eqtrd 2773 . . . . 5 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = βˆ…)
1918ex 414 . . . 4 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = βˆ…))
204, 1eqtrdi 2789 . . . . 5 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = βˆ…)
2120a1d 25 . . . 4 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = βˆ…))
2219, 21pm2.61i 182 . . 3 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = βˆ…)
2311wwlknllvtx 29100 . . . . . . 7 (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ ((π‘€β€˜0) ∈ 𝑉 ∧ (π‘€β€˜π‘) ∈ 𝑉))
24 eleq1 2822 . . . . . . . . 9 (𝐴 = (π‘€β€˜0) β†’ (𝐴 ∈ 𝑉 ↔ (π‘€β€˜0) ∈ 𝑉))
2524eqcoms 2741 . . . . . . . 8 ((π‘€β€˜0) = 𝐴 β†’ (𝐴 ∈ 𝑉 ↔ (π‘€β€˜0) ∈ 𝑉))
26 eleq1 2822 . . . . . . . . 9 (𝐡 = (π‘€β€˜π‘) β†’ (𝐡 ∈ 𝑉 ↔ (π‘€β€˜π‘) ∈ 𝑉))
2726eqcoms 2741 . . . . . . . 8 ((π‘€β€˜π‘) = 𝐡 β†’ (𝐡 ∈ 𝑉 ↔ (π‘€β€˜π‘) ∈ 𝑉))
2825, 27bi2anan9 638 . . . . . . 7 (((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡) β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ↔ ((π‘€β€˜0) ∈ 𝑉 ∧ (π‘€β€˜π‘) ∈ 𝑉)))
2923, 28syl5ibrcom 246 . . . . . 6 (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ (((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡) β†’ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)))
3029con3rr3 155 . . . . 5 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ Β¬ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)))
3130ralrimiv 3146 . . . 4 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ βˆ€π‘€ ∈ (𝑁 WWalksN 𝐺) Β¬ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡))
32 rabeq0 4385 . . . 4 ({𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)} = βˆ… ↔ βˆ€π‘€ ∈ (𝑁 WWalksN 𝐺) Β¬ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡))
3331, 32sylibr 233 . . 3 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)} = βˆ…)
3422, 33eqtr4d 2776 . 2 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)})
3512adantr 482 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝑁 WWalksNOn 𝐺) = (π‘Ž ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)}))
36 eqeq2 2745 . . . . . 6 (π‘Ž = 𝐴 β†’ ((π‘€β€˜0) = π‘Ž ↔ (π‘€β€˜0) = 𝐴))
37 eqeq2 2745 . . . . . 6 (𝑏 = 𝐡 β†’ ((π‘€β€˜π‘) = 𝑏 ↔ (π‘€β€˜π‘) = 𝐡))
3836, 37bi2anan9 638 . . . . 5 ((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) β†’ (((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏) ↔ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)))
3938rabbidv 3441 . . . 4 ((π‘Ž = 𝐴 ∧ 𝑏 = 𝐡) β†’ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)} = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)})
4039adantl 483 . . 3 ((((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) ∧ (π‘Ž = 𝐴 ∧ 𝑏 = 𝐡)) β†’ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘) = 𝑏)} = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)})
41 simprl 770 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ 𝐴 ∈ 𝑉)
42 simprr 772 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ 𝐡 ∈ 𝑉)
43 ovex 7442 . . . . 5 (𝑁 WWalksN 𝐺) ∈ V
4443rabex 5333 . . . 4 {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)} ∈ V
4544a1i 11 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)} ∈ V)
4635, 40, 41, 42, 45ovmpod 7560 . 2 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)})
4710, 34, 46ecase 1032 1 (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)}
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433  Vcvv 3475  βˆ…c0 4323  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  0cc0 11110  1c1 11111   + caddc 11113  β„•0cn0 12472  β™―chash 14290  Vtxcvtx 28256  WWalkscwwlks 29079   WWalksN cwwlksn 29080   WWalksNOn cwwlksnon 29081
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-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-wwlks 29084  df-wwlksn 29085  df-wwlksnon 29086
This theorem is referenced by:  wwlknon  29111  wwlksnonfi  29174  wpthswwlks2on  29215  clwwlknclwwlkdif  29232  clwwlknclwwlkdifnum  29233  numclwwlkqhash  29628
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