Theorem clwwlknon1loop 30415

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 782	. . . 4 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → 𝑤 = ⟨“𝑋”⟩)
2		s1cl 14639	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ⟨“𝑋”⟩ ∈ Word 𝑉)
3	2	adantr 485	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → ⟨“𝑋”⟩ ∈ Word 𝑉)
4	3	adantr 485	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑤 = ⟨“𝑋”⟩) → ⟨“𝑋”⟩ ∈ Word 𝑉)
5		eleq1 2849	. . . . . . . 8 ⊢ (𝑤 = ⟨“𝑋”⟩ → (𝑤 ∈ Word 𝑉 ↔ ⟨“𝑋”⟩ ∈ Word 𝑉))
6	5	adantl 486	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑤 = ⟨“𝑋”⟩) → (𝑤 ∈ Word 𝑉 ↔ ⟨“𝑋”⟩ ∈ Word 𝑉))
7	4, 6	mpbird 260	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑤 = ⟨“𝑋”⟩) → 𝑤 ∈ Word 𝑉)
8		simpr 489	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → {𝑋} ∈ 𝐸)
9	8	anim1ci 627	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑤 = ⟨“𝑋”⟩) → (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))
10	7, 9	jca 520	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑤 = ⟨“𝑋”⟩) → (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)))
11	10	ex 417	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → (𝑤 = ⟨“𝑋”⟩ → (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))))
12	1, 11	impbid2 229	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) ↔ 𝑤 = ⟨“𝑋”⟩))
13	12	alrimiv 1955	. 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 30414	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
18	17	eqeq1d 2763	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ((𝑋𝐶1) = {⟨“𝑋”⟩} ↔ {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)} = {⟨“𝑋”⟩}))
19	18	adantr 485	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → ((𝑋𝐶1) = {⟨“𝑋”⟩} ↔ {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)} = {⟨“𝑋”⟩}))
20		rabeqsn 4632	. . 3 ⊢ ({𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)} = {⟨“𝑋”⟩} ↔ ∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) ↔ 𝑤 = ⟨“𝑋”⟩))
21	19, 20	bitrdi 290	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → ((𝑋𝐶1) = {⟨“𝑋”⟩} ↔ ∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) ↔ 𝑤 = ⟨“𝑋”⟩)))
22	13, 21	mpbird 260	1 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → (𝑋𝐶1) = {⟨“𝑋”⟩})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1566   = wceq 1568   ∈ wcel 2141  {crab 3414  {csn 4588  ‘cfv 6536  (class class class)co 7410  1c1 11100  Word cword 14549  ⟨“cs1 14632  Vtxcvtx 29312  Edgcedg 29363  ClWWalksNOncclwwlknon 30404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-oadd 8456  df-er 8693  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-n0 12504  df-xnn0 12577  df-z 12591  df-uz 12862  df-fz 13535  df-fzo 13682  df-hash 14366  df-word 14550  df-lsw 14599  df-s1 14633  df-clwwlk 30299  df-clwwlkn 30342  df-clwwlknon 30405

This theorem is referenced by:  clwwlknon1sn  30417  clwwlknon1le1  30418