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Theorem clwwlknon1loop 29091
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 = (Vtxβ€˜πΊ)
clwwlknon1.c 𝐢 = (ClWWalksNOnβ€˜πΊ)
clwwlknon1.e 𝐸 = (Edgβ€˜πΊ)
Assertion
Ref Expression
clwwlknon1loop ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) β†’ (𝑋𝐢1) = {βŸ¨β€œπ‘‹β€βŸ©})

Proof of Theorem clwwlknon1loop
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 simprl 770 . . . 4 ((𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)) β†’ 𝑀 = βŸ¨β€œπ‘‹β€βŸ©)
2 s1cl 14499 . . . . . . . . 9 (𝑋 ∈ 𝑉 β†’ βŸ¨β€œπ‘‹β€βŸ© ∈ Word 𝑉)
32adantr 482 . . . . . . . 8 ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) β†’ βŸ¨β€œπ‘‹β€βŸ© ∈ Word 𝑉)
43adantr 482 . . . . . . 7 (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑀 = βŸ¨β€œπ‘‹β€βŸ©) β†’ βŸ¨β€œπ‘‹β€βŸ© ∈ Word 𝑉)
5 eleq1 2822 . . . . . . . 8 (𝑀 = βŸ¨β€œπ‘‹β€βŸ© β†’ (𝑀 ∈ Word 𝑉 ↔ βŸ¨β€œπ‘‹β€βŸ© ∈ Word 𝑉))
65adantl 483 . . . . . . 7 (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑀 = βŸ¨β€œπ‘‹β€βŸ©) β†’ (𝑀 ∈ Word 𝑉 ↔ βŸ¨β€œπ‘‹β€βŸ© ∈ Word 𝑉))
74, 6mpbird 257 . . . . . 6 (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑀 = βŸ¨β€œπ‘‹β€βŸ©) β†’ 𝑀 ∈ Word 𝑉)
8 simpr 486 . . . . . . 7 ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) β†’ {𝑋} ∈ 𝐸)
98anim1ci 617 . . . . . 6 (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑀 = βŸ¨β€œπ‘‹β€βŸ©) β†’ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸))
107, 9jca 513 . . . . 5 (((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) ∧ 𝑀 = βŸ¨β€œπ‘‹β€βŸ©) β†’ (𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)))
1110ex 414 . . . 4 ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) β†’ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© β†’ (𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸))))
121, 11impbid2 225 . . 3 ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) β†’ ((𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)) ↔ 𝑀 = βŸ¨β€œπ‘‹β€βŸ©))
1312alrimiv 1931 . 2 ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) β†’ βˆ€π‘€((𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)) ↔ 𝑀 = βŸ¨β€œπ‘‹β€βŸ©))
14 clwwlknon1.v . . . . . 6 𝑉 = (Vtxβ€˜πΊ)
15 clwwlknon1.c . . . . . 6 𝐢 = (ClWWalksNOnβ€˜πΊ)
16 clwwlknon1.e . . . . . 6 𝐸 = (Edgβ€˜πΊ)
1714, 15, 16clwwlknon1 29090 . . . . 5 (𝑋 ∈ 𝑉 β†’ (𝑋𝐢1) = {𝑀 ∈ Word 𝑉 ∣ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)})
1817eqeq1d 2735 . . . 4 (𝑋 ∈ 𝑉 β†’ ((𝑋𝐢1) = {βŸ¨β€œπ‘‹β€βŸ©} ↔ {𝑀 ∈ Word 𝑉 ∣ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)} = {βŸ¨β€œπ‘‹β€βŸ©}))
1918adantr 482 . . 3 ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) β†’ ((𝑋𝐢1) = {βŸ¨β€œπ‘‹β€βŸ©} ↔ {𝑀 ∈ Word 𝑉 ∣ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)} = {βŸ¨β€œπ‘‹β€βŸ©}))
20 rabeqsn 4631 . . 3 ({𝑀 ∈ Word 𝑉 ∣ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)} = {βŸ¨β€œπ‘‹β€βŸ©} ↔ βˆ€π‘€((𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)) ↔ 𝑀 = βŸ¨β€œπ‘‹β€βŸ©))
2119, 20bitrdi 287 . 2 ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) β†’ ((𝑋𝐢1) = {βŸ¨β€œπ‘‹β€βŸ©} ↔ βˆ€π‘€((𝑀 ∈ Word 𝑉 ∧ (𝑀 = βŸ¨β€œπ‘‹β€βŸ© ∧ {𝑋} ∈ 𝐸)) ↔ 𝑀 = βŸ¨β€œπ‘‹β€βŸ©)))
2213, 21mpbird 257 1 ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) β†’ (𝑋𝐢1) = {βŸ¨β€œπ‘‹β€βŸ©})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107  {crab 3406  {csn 4590  β€˜cfv 6500  (class class class)co 7361  1c1 11060  Word cword 14411  βŸ¨β€œcs1 14492  Vtxcvtx 27996  Edgcedg 28047  ClWWalksNOncclwwlknon 29080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-oadd 8420  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-n0 12422  df-xnn0 12494  df-z 12508  df-uz 12772  df-fz 13434  df-fzo 13577  df-hash 14240  df-word 14412  df-lsw 14460  df-s1 14493  df-clwwlk 28975  df-clwwlkn 29018  df-clwwlknon 29081
This theorem is referenced by:  clwwlknon1sn  29093  clwwlknon1le1  29094
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