Theorem clwwlken 30039

Description: The set of closed walks of a fixed length represented by walks (as words) and the set of closed walks (as words) of the fixed length are equinumerous. (Contributed by AV, 5-Jun-2022.) (Proof shortened by AV, 2-Nov-2022.)

Assertion

Ref	Expression
clwwlken	⊢ (𝑁 ∈ ℕ → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} ≈ (𝑁 ClWWalksN 𝐺))

Distinct variable groups:   𝑤,𝐺   𝑤,𝑁

Proof of Theorem clwwlken

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7385	. . 3 ⊢ (𝑁 WWalksN 𝐺) ∈ V
2	1	rabex 5279	. 2 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} ∈ V
3		ovex 7385	. 2 ⊢ (𝑁 ClWWalksN 𝐺) ∈ V
4		eqid 2731	. . 3 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}
5		eqid 2731	. . 3 ⊢ (𝑐 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} ↦ (𝑐 prefix 𝑁)) = (𝑐 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} ↦ (𝑐 prefix 𝑁))
6	4, 5	clwwlkf1o 30038	. 2 ⊢ (𝑁 ∈ ℕ → (𝑐 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} ↦ (𝑐 prefix 𝑁)):{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}–1-1-onto→(𝑁 ClWWalksN 𝐺))
7		f1oen2g 8897	. 2 ⊢ (({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} ∈ V ∧ (𝑁 ClWWalksN 𝐺) ∈ V ∧ (𝑐 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} ↦ (𝑐 prefix 𝑁)):{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}–1-1-onto→(𝑁 ClWWalksN 𝐺)) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} ≈ (𝑁 ClWWalksN 𝐺))
8	2, 3, 6, 7	mp3an12i 1467	1 ⊢ (𝑁 ∈ ℕ → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} ≈ (𝑁 ClWWalksN 𝐺))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436   class class class wbr 5093   ↦ cmpt 5174  –1-1-onto→wf1o 6486  ‘cfv 6487  (class class class)co 7352   ≈ cen 8872  0cc0 11012  ℕcn 12131  lastSclsw 14475   prefix cpfx 14584   WWalksN cwwlksn 29811   ClWWalksN cclwwlkn 30011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11068  ax-resscn 11069  ax-1cn 11070  ax-icn 11071  ax-addcl 11072  ax-addrcl 11073  ax-mulcl 11074  ax-mulrcl 11075  ax-mulcom 11076  ax-addass 11077  ax-mulass 11078  ax-distr 11079  ax-i2m1 11080  ax-1ne0 11081  ax-1rid 11082  ax-rnegex 11083  ax-rrecex 11084  ax-cnre 11085  ax-pre-lttri 11086  ax-pre-lttrn 11087  ax-pre-ltadd 11088  ax-pre-mulgt0 11089

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-oadd 8395  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-card 9838  df-pnf 11154  df-mnf 11155  df-xr 11156  df-ltxr 11157  df-le 11158  df-sub 11352  df-neg 11353  df-nn 12132  df-n0 12388  df-xnn0 12461  df-z 12475  df-uz 12739  df-rp 12897  df-fz 13414  df-fzo 13561  df-hash 14244  df-word 14427  df-lsw 14476  df-concat 14484  df-s1 14510  df-substr 14555  df-pfx 14585  df-wwlks 29815  df-wwlksn 29816  df-clwwlk 29969  df-clwwlkn 30012

This theorem is referenced by: (None)