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Theorem clwlknf1oclwwlknlem2 30026

Description: Lemma 2 for clwlknf1oclwwlkn 30028: The closed walks of a positive length are nonempty closed walks of this length. (Contributed by AV, 26-May-2022.)

**Assertion**
Ref	Expression
clwlknf1oclwwlknlem2	⊢ (𝑁 ∈ ℕ → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)})

Distinct variable groups: 𝐺,𝑐,𝑤 𝑁,𝑐,𝑤

**Proof of Theorem clwlknf1oclwwlknlem2**
Step	Hyp	Ref	Expression
1		2fveq3 6827	. . . 4 ⊢ (𝑤 = 𝑐 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑐)))
2	1	eqeq1d 2731	. . 3 ⊢ (𝑤 = 𝑐 → ((♯‘(1^st ‘𝑤)) = 𝑁 ↔ (♯‘(1^st ‘𝑐)) = 𝑁))
3	2	cbvrabv 3405	. 2 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑐)) = 𝑁}
4		nnge1 12156	. . . . 5 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
5		breq2 5096	. . . . 5 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → (1 ≤ (♯‘(1^st ‘𝑐)) ↔ 1 ≤ 𝑁))
6	4, 5	syl5ibrcom 247	. . . 4 ⊢ (𝑁 ∈ ℕ → ((♯‘(1^st ‘𝑐)) = 𝑁 → 1 ≤ (♯‘(1^st ‘𝑐))))
7	6	pm4.71rd 562	. . 3 ⊢ (𝑁 ∈ ℕ → ((♯‘(1^st ‘𝑐)) = 𝑁 ↔ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)))
8	7	rabbidv 3402	. 2 ⊢ (𝑁 ∈ ℕ → {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑐)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)})
9	3, 8	eqtrid 2776	1 ⊢ (𝑁 ∈ ℕ → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {crab 3394 class class class wbr 5092 ‘cfv 6482 1^st c1st 7922 1c1 11010 ≤ cle 11150 ℕcn 12128 ♯chash 14237 ClWalkscclwlks 29715

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 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 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129

This theorem is referenced by: clwlknf1oclwwlkn 30028

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