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Theorem clwlknf1oclwwlknlem3OLD 27417
Description: Obsolete version of clwlknf1oclwwlknlem3 27415 as of 12-Oct-2022. (Contributed by AV, 26-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
clwlknf1oclwwlknOLD.a 𝐴 = (1st𝑐)
clwlknf1oclwwlknOLD.b 𝐵 = (2nd𝑐)
clwlknf1oclwwlknOLD.c 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}
clwlknf1oclwwlknOLD.f 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))
Assertion
Ref Expression
clwlknf1oclwwlknlem3OLD ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩)) ↾ 𝐶))
Distinct variable groups:   𝐶,𝑐   𝐺,𝑐,𝑤   𝑤,𝑁
Allowed substitution hints:   𝐴(𝑤,𝑐)   𝐵(𝑤,𝑐)   𝐶(𝑤)   𝐹(𝑤,𝑐)   𝑁(𝑐)

Proof of Theorem clwlknf1oclwwlknlem3OLD
StepHypRef Expression
1 clwlknf1oclwwlknOLD.c . . . 4 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}
2 nnge1 11342 . . . . . . 7 (𝑁 ∈ ℕ → 1 ≤ 𝑁)
3 breq2 4847 . . . . . . 7 ((♯‘(1st𝑤)) = 𝑁 → (1 ≤ (♯‘(1st𝑤)) ↔ 1 ≤ 𝑁))
42, 3syl5ibrcom 239 . . . . . 6 (𝑁 ∈ ℕ → ((♯‘(1st𝑤)) = 𝑁 → 1 ≤ (♯‘(1st𝑤))))
54ad2antlr 719 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑤 ∈ (ClWalks‘𝐺)) → ((♯‘(1st𝑤)) = 𝑁 → 1 ≤ (♯‘(1st𝑤))))
65ss2rabdv 3879 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ⊆ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))})
71, 6syl5eqss 3845 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 ⊆ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))})
8 resmpt 5661 . . 3 (𝐶 ⊆ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩)) ↾ 𝐶) = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩)))
97, 8syl 17 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩)) ↾ 𝐶) = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩)))
10 clwlknf1oclwwlknOLD.f . 2 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))
119, 10syl6reqr 2852 1 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩)) ↾ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  {crab 3093  wss 3769  cop 4374   class class class wbr 4843  cmpt 4922  cres 5314  cfv 6101  (class class class)co 6878  1st c1st 7399  2nd c2nd 7400  0cc0 10224  1c1 10225  cle 10364  cn 11312  chash 13370   substr csubstr 13664  USPGraphcuspgr 26384  ClWalkscclwlks 27024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313
This theorem is referenced by:  clwlknf1oclwwlknOLD  27418
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