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Theorem clwlknf1oclwwlknlem3 29604

Description: Lemma 3 for clwlknf1oclwwlkn 29605: The bijective function of clwlknf1oclwwlkn 29605 is the bijective function of clwlkclwwlkf1o 29532 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 = (1^st ‘𝑐)
clwlknf1oclwwlkn.b	⊢ 𝐵 = (2^nd ‘𝑐)
clwlknf1oclwwlkn.c	⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁}
clwlknf1oclwwlkn.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 prefix (♯‘𝐴)))

**Assertion**
Ref	Expression
clwlknf1oclwwlknlem3	⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘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) ∣ (♯‘(1^st ‘𝑤)) = 𝑁}
3		nnge1 12245	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
4		breq2 5152	. . . . . . 7 ⊢ ((♯‘(1^st ‘𝑤)) = 𝑁 → (1 ≤ (♯‘(1^st ‘𝑤)) ↔ 1 ≤ 𝑁))
5	3, 4	syl5ibrcom 246	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ((♯‘(1^st ‘𝑤)) = 𝑁 → 1 ≤ (♯‘(1^st ‘𝑤))))
6	5	ad2antlr 724	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑤 ∈ (ClWalks‘𝐺)) → ((♯‘(1^st ‘𝑤)) = 𝑁 → 1 ≤ (♯‘(1^st ‘𝑤))))
7	6	ss2rabdv 4073	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} ⊆ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))})
8	2, 7	eqsstrid 4030	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 ⊆ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))})
9	8	resmptd 6040	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶) = (𝑐 ∈ 𝐶 ↦ (𝐵 prefix (♯‘𝐴))))
10	1, 9	eqtr4id 2790	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 {crab 3431 class class class wbr 5148 ↦ cmpt 5231 ↾ cres 5678 ‘cfv 6543 (class class class)co 7412 1^st c1st 7977 2^nd c2nd 7978 1c1 11115 ≤ cle 11254 ℕcn 12217 ♯chash 14295 prefix cpfx 14625 USPGraphcuspgr 28676 ClWalkscclwlks 29295

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218

This theorem is referenced by: clwlknf1oclwwlkn 29605

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