Theorem clwwlknon1loop 30027

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 14567	. . . . . . . . 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 2816	. . . . . . . 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 1927	. 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 30026	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
18	17	eqeq1d 2731	. . . 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 4631	. . 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 1538   = wceq 1540   ∈ wcel 2109  {crab 3405  {csn 4589  ‘cfv 6511  (class class class)co 7387  1c1 11069  Word cword 14478  ⟨“cs1 14560  Vtxcvtx 28923  Edgcedg 28974  ClWWalksNOncclwwlknon 30016

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-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-oadd 8438  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-n0 12443  df-xnn0 12516  df-z 12530  df-uz 12794  df-fz 13469  df-fzo 13616  df-hash 14296  df-word 14479  df-lsw 14528  df-s1 14561  df-clwwlk 29911  df-clwwlkn 29954  df-clwwlknon 30017

This theorem is referenced by:  clwwlknon1sn  30029  clwwlknon1le1  30030