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Theorem iswwlksnon 28798
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 7394 . . 3 (π΄βˆ…π΅) = βˆ…
2 df-wwlksnon 28777 . . . . 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 7594 . . . 4 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝑁 WWalksNOn 𝐺) = βˆ…)
43oveqd 7374 . . 3 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = (π΄βˆ…π΅))
5 df-wwlksn 28776 . . . . . 6 WWalksN = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ {𝑀 ∈ (WWalksβ€˜π‘”) ∣ (β™―β€˜π‘€) = (𝑛 + 1)})
65mpondm0 7594 . . . . 5 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (𝑁 WWalksN 𝐺) = βˆ…)
76rabeqdv 3422 . . . 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 4342 . . . 4 {𝑀 ∈ βˆ… ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)} = βˆ…
97, 8eqtrdi 2792 . . 3 (Β¬ (𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡)} = βˆ…)
101, 4, 93eqtr4a 2802 . 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 28796 . . . . . . . 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 481 . . . . . . 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 7374 . . . . . 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 2736 . . . . . . . 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 7594 . . . . . . 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 482 . . . . . 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 2776 . . . . 5 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = βˆ…)
1918ex 413 . . . 4 ((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) β†’ (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(𝑁 WWalksNOn 𝐺)𝐡) = βˆ…))
204, 1eqtrdi 2792 . . . . 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 28791 . . . . . . 7 (𝑀 ∈ (𝑁 WWalksN 𝐺) β†’ ((π‘€β€˜0) ∈ 𝑉 ∧ (π‘€β€˜π‘) ∈ 𝑉))
24 eleq1 2825 . . . . . . . . 9 (𝐴 = (π‘€β€˜0) β†’ (𝐴 ∈ 𝑉 ↔ (π‘€β€˜0) ∈ 𝑉))
2524eqcoms 2744 . . . . . . . 8 ((π‘€β€˜0) = 𝐴 β†’ (𝐴 ∈ 𝑉 ↔ (π‘€β€˜0) ∈ 𝑉))
26 eleq1 2825 . . . . . . . . 9 (𝐡 = (π‘€β€˜π‘) β†’ (𝐡 ∈ 𝑉 ↔ (π‘€β€˜π‘) ∈ 𝑉))
2726eqcoms 2744 . . . . . . . 8 ((π‘€β€˜π‘) = 𝐡 β†’ (𝐡 ∈ 𝑉 ↔ (π‘€β€˜π‘) ∈ 𝑉))
2825, 27bi2anan9 637 . . . . . . 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 3142 . . . 4 (Β¬ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ βˆ€π‘€ ∈ (𝑁 WWalksN 𝐺) Β¬ ((π‘€β€˜0) = 𝐴 ∧ (π‘€β€˜π‘) = 𝐡))
32 rabeq0 4344 . . . 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 2779 . 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 481 . . 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 2748 . . . . . 6 (π‘Ž = 𝐴 β†’ ((π‘€β€˜0) = π‘Ž ↔ (π‘€β€˜0) = 𝐴))
37 eqeq2 2748 . . . . . 6 (𝑏 = 𝐡 β†’ ((π‘€β€˜π‘) = 𝑏 ↔ (π‘€β€˜π‘) = 𝐡))
3836, 37bi2anan9 637 . . . . 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 3415 . . . 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 482 . . 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 769 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ 𝐴 ∈ 𝑉)
42 simprr 771 . . 3 (((𝑁 ∈ β„•0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉)) β†’ 𝐡 ∈ 𝑉)
43 ovex 7390 . . . . 5 (𝑁 WWalksN 𝐺) ∈ V
4443rabex 5289 . . . 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 7507 . 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 1031 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 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3064  {crab 3407  Vcvv 3445  βˆ…c0 4282  β€˜cfv 6496  (class class class)co 7357   ∈ cmpo 7359  0cc0 11051  1c1 11052   + caddc 11054  β„•0cn0 12413  β™―chash 14230  Vtxcvtx 27947  WWalkscwwlks 28770   WWalksN cwwlksn 28771   WWalksNOn cwwlksnon 28772
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-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-wwlks 28775  df-wwlksn 28776  df-wwlksnon 28777
This theorem is referenced by:  wwlknon  28802  wwlksnonfi  28865  wpthswwlks2on  28906  clwwlknclwwlkdif  28923  clwwlknclwwlkdifnum  28924  numclwwlkqhash  29319
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