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Theorem clwlknf1oclwwlknlem2 27405

Description: Lemma 2 for clwlknf1oclwwlkn 27407: 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 6414	. . . 4 ⊢ (𝑤 = 𝑐 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑐)))
2	1	eqeq1d 2799	. . 3 ⊢ (𝑤 = 𝑐 → ((♯‘(1^st ‘𝑤)) = 𝑁 ↔ (♯‘(1^st ‘𝑐)) = 𝑁))
3	2	cbvrabv 3381	. 2 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑐)) = 𝑁}
4		nnge1 11340	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
5		breq2 4845	. . . . . 6 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → (1 ≤ (♯‘(1^st ‘𝑐)) ↔ 1 ≤ 𝑁))
6	4, 5	syl5ibrcom 239	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((♯‘(1^st ‘𝑐)) = 𝑁 → 1 ≤ (♯‘(1^st ‘𝑐))))
7	6	pm4.71rd 559	. . . 4 ⊢ (𝑁 ∈ ℕ → ((♯‘(1^st ‘𝑐)) = 𝑁 ↔ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)))
8	7	adantr 473	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑐 ∈ (ClWalks‘𝐺)) → ((♯‘(1^st ‘𝑐)) = 𝑁 ↔ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)))
9	8	rabbidva 3370	. 2 ⊢ (𝑁 ∈ ℕ → {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑐)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)})
10	3, 9	syl5eq 2843	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 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 {crab 3091 class class class wbr 4841 ‘cfv 6099 1^st c1st 7397 1c1 10223 ≤ cle 10362 ℕcn 11310 ♯chash 13366 ClWalkscclwlks 27015

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 2375 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299

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 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311

This theorem is referenced by: clwlknf1oclwwlkn 27407 clwlknf1oclwwlknOLD 27409

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