Theorem eclclwwlkn1 28484

Description: An equivalence class according to ∼. (Contributed by Alexander van der Vekens, 12-Apr-2018.) (Revised by AV, 30-Apr-2021.)

Hypotheses

Ref	Expression
erclwwlkn.w	⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)
erclwwlkn.r	⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}

Assertion

Ref	Expression
eclclwwlkn1	⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))

Distinct variable groups:   𝑡,𝑊,𝑢   𝑛,𝑁,𝑢,𝑡,𝑥,𝑦   𝑛,𝑊   𝑥, ∼ ,𝑦   𝑥,𝑊   𝑥,𝐺   𝑥,𝑋   𝑥,𝐵,𝑦   𝑦,𝑁   𝑦,𝑊   𝑦,𝑋

Allowed substitution hints:   𝐵(𝑢,𝑡,𝑛)   ∼ (𝑢,𝑡,𝑛)   𝐺(𝑦,𝑢,𝑡,𝑛)   𝑋(𝑢,𝑡,𝑛)

Proof of Theorem eclclwwlkn1

Step	Hyp	Ref	Expression
1		elqsecl 8591	. 2 ⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∣ 𝑥 ∼ 𝑦}))
2		erclwwlkn.w	. . . . . . . . 9 ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)
3		erclwwlkn.r	. . . . . . . . 9 ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
4	2, 3	erclwwlknsym 28479	. . . . . . . 8 ⊢ (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥)
5	2, 3	erclwwlknsym 28479	. . . . . . . 8 ⊢ (𝑦 ∼ 𝑥 → 𝑥 ∼ 𝑦)
6	4, 5	impbii 208	. . . . . . 7 ⊢ (𝑥 ∼ 𝑦 ↔ 𝑦 ∼ 𝑥)
7	6	a1i 11	. . . . . 6 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → (𝑥 ∼ 𝑦 ↔ 𝑦 ∼ 𝑥))
8	7	abbidv 2805	. . . . 5 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → {𝑦 ∣ 𝑥 ∼ 𝑦} = {𝑦 ∣ 𝑦 ∼ 𝑥})
9	2, 3	erclwwlkneq 28476	. . . . . . . 8 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
10	9	el2v 3445	. . . . . . 7 ⊢ (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)))
11	10	a1i 11	. . . . . 6 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → (𝑦 ∼ 𝑥 ↔ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
12	11	abbidv 2805	. . . . 5 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → {𝑦 ∣ 𝑦 ∼ 𝑥} = {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))})
13		3anan12 1096	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)) ↔ (𝑥 ∈ 𝑊 ∧ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
14		ibar 530	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑊 → ((𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)) ↔ (𝑥 ∈ 𝑊 ∧ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)))))
15	14	bicomd 222	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑊 → ((𝑥 ∈ 𝑊 ∧ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))) ↔ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
16	15	adantl 483	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → ((𝑥 ∈ 𝑊 ∧ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))) ↔ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
17	13, 16	bitrid 283	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → ((𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)) ↔ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))))
18	17	abbidv 2805	. . . . . 6 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))} = {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))})
19		df-rab 3287	. . . . . 6 ⊢ {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)} = {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))}
20	18, 19	eqtr4di 2794	. . . . 5 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛))} = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
21	8, 12, 20	3eqtrd 2780	. . . 4 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → {𝑦 ∣ 𝑥 ∼ 𝑦} = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})
22	21	eqeq2d 2747	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝑥 ∈ 𝑊) → (𝐵 = {𝑦 ∣ 𝑥 ∼ 𝑦} ↔ 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
23	22	rexbidva 3170	. 2 ⊢ (𝐵 ∈ 𝑋 → (∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∣ 𝑥 ∼ 𝑦} ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
24	1, 23	bitrd 279	1 ⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104  {cab 2713  ∃wrex 3071  {crab 3284  Vcvv 3437   class class class wbr 5081  {copab 5143  (class class class)co 7307   / cqs 8528  0cc0 10917  ...cfz 13285   cyclShift ccsh 14546   ClWWalksN cclwwlkn 28433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-cnex 10973  ax-resscn 10974  ax-1cn 10975  ax-icn 10976  ax-addcl 10977  ax-addrcl 10978  ax-mulcl 10979  ax-mulrcl 10980  ax-mulcom 10981  ax-addass 10982  ax-mulass 10983  ax-distr 10984  ax-i2m1 10985  ax-1ne0 10986  ax-1rid 10987  ax-rnegex 10988  ax-rrecex 10989  ax-cnre 10990  ax-pre-lttri 10991  ax-pre-lttrn 10992  ax-pre-ltadd 10993  ax-pre-mulgt0 10994  ax-pre-sup 10995

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-om 7745  df-1st 7863  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-1o 8328  df-er 8529  df-ec 8531  df-qs 8535  df-map 8648  df-en 8765  df-dom 8766  df-sdom 8767  df-fin 8768  df-sup 9245  df-inf 9246  df-card 9741  df-pnf 11057  df-mnf 11058  df-xr 11059  df-ltxr 11060  df-le 11061  df-sub 11253  df-neg 11254  df-div 11679  df-nn 12020  df-2 12082  df-n0 12280  df-z 12366  df-uz 12629  df-rp 12777  df-fz 13286  df-fzo 13429  df-fl 13558  df-mod 13636  df-hash 14091  df-word 14263  df-concat 14319  df-substr 14399  df-pfx 14429  df-csh 14547  df-clwwlk 28391  df-clwwlkn 28434

This theorem is referenced by:  eleclclwwlkn  28485  hashecclwwlkn1  28486  umgrhashecclwwlk  28487