Theorem clwlknf1oclwwlknlem3 28447

Description: Lemma 3 for clwlknf1oclwwlkn 28448: The bijective function of clwlknf1oclwwlkn 28448 is the bijective function of clwlkclwwlkf1o 28375 restricted to the closed walks with a fixed positive length. (Contributed by AV, 26-May-2022.) (Revised by AV, 1-Nov-2022.)

Hypotheses

Ref	Expression
clwlknf1oclwwlkn.a	⊢ 𝐴 = (1st ‘𝑐)
clwlknf1oclwwlkn.b	⊢ 𝐵 = (2nd ‘𝑐)
clwlknf1oclwwlkn.c	⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁}
clwlknf1oclwwlkn.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 prefix (♯‘𝐴)))

Assertion

Ref	Expression
clwlknf1oclwwlknlem3	⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ↦ (𝐵 prefix (♯‘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 clwlknf1oclwwlknlem3

Step	Hyp	Ref	Expression
1		clwlknf1oclwwlkn.f	. 2 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 prefix (♯‘𝐴)))
2		clwlknf1oclwwlkn.c	. . . 4 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁}
3		nnge1 12001	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
4		breq2 5078	. . . . . . 7 ⊢ ((♯‘(1st ‘𝑤)) = 𝑁 → (1 ≤ (♯‘(1st ‘𝑤)) ↔ 1 ≤ 𝑁))
5	3, 4	syl5ibrcom 246	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((♯‘(1st ‘𝑤)) = 𝑁 → 1 ≤ (♯‘(1st ‘𝑤))))
6	5	ad2antlr 724	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑤 ∈ (ClWalks‘𝐺)) → ((♯‘(1st ‘𝑤)) = 𝑁 → 1 ≤ (♯‘(1st ‘𝑤))))
7	6	ss2rabdv 4009	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁} ⊆ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))})
8	2, 7	eqsstrid 3969	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 ⊆ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))})
9	8	resmptd 5948	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶) = (𝑐 ∈ 𝐶 ↦ (𝐵 prefix (♯‘𝐴))))
10	1, 9	eqtr4id 2797	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {crab 3068   class class class wbr 5074   ↦ cmpt 5157   ↾ cres 5591  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  1c1 10872   ≤ cle 11010  ℕcn 11973  ♯chash 14044   prefix cpfx 14383  USPGraphcuspgr 27518  ClWalkscclwlks 28138

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974

This theorem is referenced by:  clwlknf1oclwwlkn  28448