Theorem clwwlknon1loop 30080

Description: If there is a loop at vertex 𝑋, the set of (closed) walks on 𝑋 of length 1 as words over the set of vertices is a singleton containing the singleton word consisting of 𝑋. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 25-Mar-2022.)

Hypotheses

Ref	Expression
clwwlknon1.v	⊢ 𝑉 = (Vtx‘𝐺)
clwwlknon1.c	⊢ 𝐶 = (ClWWalksNOn‘𝐺)
clwwlknon1.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
clwwlknon1loop	⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → (𝑋𝐶1) = {⟨“𝑋”⟩})

Proof of Theorem clwwlknon1loop

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 770	. . . 4 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → 𝑤 = ⟨“𝑋”⟩)
2		s1cl 14512	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ⟨“𝑋”⟩ ∈ Word 𝑉)
3	2	adantr 480	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → ⟨“𝑋”⟩ ∈ Word 𝑉)
4	3	adantr 480	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑤 = ⟨“𝑋”⟩) → ⟨“𝑋”⟩ ∈ Word 𝑉)
5		eleq1 2821	. . . . . . . 8 ⊢ (𝑤 = ⟨“𝑋”⟩ → (𝑤 ∈ Word 𝑉 ↔ ⟨“𝑋”⟩ ∈ Word 𝑉))
6	5	adantl 481	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑤 = ⟨“𝑋”⟩) → (𝑤 ∈ Word 𝑉 ↔ ⟨“𝑋”⟩ ∈ Word 𝑉))
7	4, 6	mpbird 257	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑤 = ⟨“𝑋”⟩) → 𝑤 ∈ Word 𝑉)
8		simpr 484	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → {𝑋} ∈ 𝐸)
9	8	anim1ci 616	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑤 = ⟨“𝑋”⟩) → (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))
10	7, 9	jca 511	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑤 = ⟨“𝑋”⟩) → (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)))
11	10	ex 412	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → (𝑤 = ⟨“𝑋”⟩ → (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))))
12	1, 11	impbid2 226	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) ↔ 𝑤 = ⟨“𝑋”⟩))
13	12	alrimiv 1928	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → ∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) ↔ 𝑤 = ⟨“𝑋”⟩))
14		clwwlknon1.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
15		clwwlknon1.c	. . . . . 6 ⊢ 𝐶 = (ClWWalksNOn‘𝐺)
16		clwwlknon1.e	. . . . . 6 ⊢ 𝐸 = (Edg‘𝐺)
17	14, 15, 16	clwwlknon1 30079	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
18	17	eqeq1d 2735	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ((𝑋𝐶1) = {⟨“𝑋”⟩} ↔ {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)} = {⟨“𝑋”⟩}))
19	18	adantr 480	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → ((𝑋𝐶1) = {⟨“𝑋”⟩} ↔ {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)} = {⟨“𝑋”⟩}))
20		rabeqsn 4619	. . 3 ⊢ ({𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)} = {⟨“𝑋”⟩} ↔ ∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) ↔ 𝑤 = ⟨“𝑋”⟩))
21	19, 20	bitrdi 287	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → ((𝑋𝐶1) = {⟨“𝑋”⟩} ↔ ∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) ↔ 𝑤 = ⟨“𝑋”⟩)))
22	13, 21	mpbird 257	1 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → (𝑋𝐶1) = {⟨“𝑋”⟩})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2113  {crab 3396  {csn 4575  ‘cfv 6486  (class class class)co 7352  1c1 11014  Word cword 14422  ⟨“cs1 14505  Vtxcvtx 28976  Edgcedg 29027  ClWWalksNOncclwwlknon 30069

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  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 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-oadd 8395  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-n0 12389  df-xnn0 12462  df-z 12476  df-uz 12739  df-fz 13410  df-fzo 13557  df-hash 14240  df-word 14423  df-lsw 14472  df-s1 14506  df-clwwlk 29964  df-clwwlkn 30007  df-clwwlknon 30070

This theorem is referenced by:  clwwlknon1sn  30082  clwwlknon1le1  30083