Theorem clwwlknon1loop 30034

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 14577	. . . . . . . . 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 2817	. . . . . . . 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 30033	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
18	17	eqeq1d 2732	. . . 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 4639	. . 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 3411  {csn 4597  ‘cfv 6519  (class class class)co 7394  1c1 11087  Word cword 14488  ⟨“cs1 14570  Vtxcvtx 28930  Edgcedg 28981  ClWWalksNOncclwwlknon 30023

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 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718  ax-cnex 11142  ax-resscn 11143  ax-1cn 11144  ax-icn 11145  ax-addcl 11146  ax-addrcl 11147  ax-mulcl 11148  ax-mulrcl 11149  ax-mulcom 11150  ax-addass 11151  ax-mulass 11152  ax-distr 11153  ax-i2m1 11154  ax-1ne0 11155  ax-1rid 11156  ax-rnegex 11157  ax-rrecex 11158  ax-cnre 11159  ax-pre-lttri 11160  ax-pre-lttrn 11161  ax-pre-ltadd 11162  ax-pre-mulgt0 11163

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-nel 3032  df-ral 3047  df-rex 3056  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-pss 3942  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-int 4919  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5541  df-eprel 5546  df-po 5554  df-so 5555  df-fr 5599  df-we 5601  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-riota 7351  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7851  df-1st 7977  df-2nd 7978  df-frecs 8269  df-wrecs 8300  df-recs 8349  df-rdg 8387  df-1o 8443  df-oadd 8447  df-er 8682  df-map 8805  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-card 9910  df-pnf 11228  df-mnf 11229  df-xr 11230  df-ltxr 11231  df-le 11232  df-sub 11425  df-neg 11426  df-nn 12198  df-n0 12459  df-xnn0 12532  df-z 12546  df-uz 12810  df-fz 13482  df-fzo 13629  df-hash 14306  df-word 14489  df-lsw 14538  df-s1 14571  df-clwwlk 29918  df-clwwlkn 29961  df-clwwlknon 30024

This theorem is referenced by:  clwwlknon1sn  30036  clwwlknon1le1  30037